The document provides an introduction to linear algebra concepts for machine learning. It defines vectors as ordered tuples of numbers that express magnitude and direction. Vector spaces are sets that contain all linear combinations of vectors. Linear independence and basis of vector spaces are discussed. Norms measure the magnitude of a vector, with examples given of the 1-norm and 2-norm. Inner products measure the correlation between vectors. Matrices can represent linear operators between vector spaces. Key linear algebra concepts such as trace, determinant, and matrix decompositions are outlined for machine learning applications.
1. Linear Algebra for Machine Learning: Linear SystemsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the first part which is giving a short overview of matrices and discussing linear systems.
3. Linear Algebra for Machine Learning: Factorization and Linear TransformationsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the third part which is discussing factorization and linear transformations.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Analysis of data is an important task in data managements systems. Many mathematical tools are used in data analysis. A new division of data management has appeared in machine learning, linear algebra, an optimal tool to analyse and manipulate the data. Data science is a multi-disciplinary subject that uses scientific methods to process the structured and unstructured data to extract the knowledge by applying suitable algorithms and systems. The strength of linear algebra is ignored by the researchers due to the poor understanding. It powers major areas of Data Science including the hot fields of Natural Language Processing and Computer Vision. The data science enthusiasts finding the programming languages for data science are easy to analyze the big data rather than using mathematical tools like linear algebra. Linear algebra is a must-know subject in data science. It will open up possibilities of working and manipulating data. In this paper, some applications of Linear Algebra in Data Science are explained.
1. Linear Algebra for Machine Learning: Linear SystemsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the first part which is giving a short overview of matrices and discussing linear systems.
3. Linear Algebra for Machine Learning: Factorization and Linear TransformationsCeni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the third part which is discussing factorization and linear transformations.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Analysis of data is an important task in data managements systems. Many mathematical tools are used in data analysis. A new division of data management has appeared in machine learning, linear algebra, an optimal tool to analyse and manipulate the data. Data science is a multi-disciplinary subject that uses scientific methods to process the structured and unstructured data to extract the knowledge by applying suitable algorithms and systems. The strength of linear algebra is ignored by the researchers due to the poor understanding. It powers major areas of Data Science including the hot fields of Natural Language Processing and Computer Vision. The data science enthusiasts finding the programming languages for data science are easy to analyze the big data rather than using mathematical tools like linear algebra. Linear algebra is a must-know subject in data science. It will open up possibilities of working and manipulating data. In this paper, some applications of Linear Algebra in Data Science are explained.
Logistic Regression in Python | Logistic Regression Example | Machine Learnin...Edureka!
** Python Data Science Training : https://www.edureka.co/python **
This Edureka Video on Logistic Regression in Python will give you basic understanding of Logistic Regression Machine Learning Algorithm with examples. In this video, you will also get to see demo on Logistic Regression using Python. Below are the topics covered in this tutorial:
1. What is Regression?
2. What is Logistic Regression?
3. Why use Logistic Regression?
4. Linear vs Logistic Regression
5. Logistic Regression Use Cases
6. Logistic Regression Example Demo in Python
Subscribe to our channel to get video updates. Hit the subscribe button above.
Machine Learning Tutorial Playlist: https://goo.gl/UxjTxm
k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells.
Machine Learning Tutorial Part - 1 | Machine Learning Tutorial For Beginners ...Simplilearn
This presentation on Machine Learning will help you understand why Machine Learning came into picture, what is Machine Learning, types of Machine Learning, Machine Learning algorithms with a detailed explanation on linear regression, decision tree & support vector machine and at the end you will also see a use case implementation where we classify whether a recipe is of a cupcake or muffin using SVM algorithm. Machine learning is a core sub-area of artificial intelligence; it enables computers to get into a mode of self-learning without being explicitly programmed. When exposed to new data, these computer programs are enabled to learn, grow, change, and develop by themselves. So, to put simply, the iterative aspect of machine learning is the ability to adapt to new data independently. Now, let us get started with this Machine Learning presentation and understand what it is and why it matters.
Below topics are explained in this Machine Learning presentation:
1. Why Machine Learning?
2. What is Machine Learning?
3. Types of Machine Learning
4. Machine Learning Algorithms
- Linear Regression
- Decision Trees
- Support Vector Machine
5. Use case: Classify whether a recipe is of a cupcake or a muffin using SVM
About Simplilearn Machine Learning course:
A form of artificial intelligence, Machine Learning is revolutionizing the world of computing as well as all people’s digital interactions. Machine Learning powers such innovative automated technologies as recommendation engines, facial recognition, fraud protection and even self-driving cars. This Machine Learning course prepares engineers, data scientists and other professionals with knowledge and hands-on skills required for certification and job competency in Machine Learning.
Why learn Machine Learning?
Machine Learning is taking over the world- and with that, there is a growing need among companies for professionals to know the ins and outs of Machine Learning
The Machine Learning market size is expected to grow from USD 1.03 Billion in 2016 to USD 8.81 Billion by 2022, at a Compound Annual Growth Rate (CAGR) of 44.1% during the forecast period.
We recommend this Machine Learning training course for the following professionals in particular:
1. Developers aspiring to be a data scientist or Machine Learning engineer
2. Information architects who want to gain expertise in Machine Learning algorithms
3. Analytics professionals who want to work in Machine Learning or artificial intelligence
4. Graduates looking to build a career in data science and Machine Learning
Learn more at: https://www.simplilearn.com/
Linear Regression vs Logistic Regression | EdurekaEdureka!
YouTube: https://youtu.be/OCwZyYH14uw
** Data Science Certification using R: https://www.edureka.co/data-science **
This Edureka PPT on Linear Regression Vs Logistic Regression covers the basic concepts of linear and logistic models. The following topics are covered in this session:
Types of Machine Learning
Regression Vs Classification
What is Linear Regression?
What is Logistic Regression?
Linear Regression Use Case
Logistic Regression Use Case
Linear Regression Vs Logistic Regression
Blog Series: http://bit.ly/data-science-blogs
Data Science Training Playlist: http://bit.ly/data-science-playlist
Follow us to never miss an update in the future.
YouTube: https://www.youtube.com/user/edurekaIN
Instagram: https://www.instagram.com/edureka_learning/
Facebook: https://www.facebook.com/edurekaIN/
Twitter: https://twitter.com/edurekain
LinkedIn: https://www.linkedin.com/company/edureka
In machine learning, support-vector machines (SVMs, also support-vector networks) are supervised learning models with associated learning algorithms that analyze data for classification and regression analysis
Convolutional Neural Network - CNN | How CNN Works | Deep Learning Course | S...Simplilearn
This presentation on Convolutional neural network tutorial (CNN) will help you understand what is a convolutional neural network, hoe CNN recognizes images, what are layers in the convolutional neural network and at the end, you will see a use case implementation using CNN. CNN is a feed forward neural network that is generally used to analyze visual images by processing data with grid like topology. A CNN is also known as a "ConvNet". Convolutional networks can also perform optical character recognition to digitize text and make natural-language processing possible on analog and hand-written documents. CNNs can also be applied to sound when it is represented visually as a spectrogram. Now, lets deep dive into this presentation to understand what is CNN and how do they actually work.
Below topics are explained in this CNN presentation(Convolutional Neural Network presentation)
1. Introduction to CNN
2. What is a convolutional neural network?
3. How CNN recognizes images?
4. Layers in convolutional neural network
5. Use case implementation using CNN
Simplilearn’s Deep Learning course will transform you into an expert in deep learning techniques using TensorFlow, the open-source software library designed to conduct machine learning & deep neural network research. With our deep learning course, you’ll master deep learning and TensorFlow concepts, learn to implement algorithms, build artificial neural networks and traverse layers of data abstraction to understand the power of data and prepare you for your new role as deep learning scientist.
Why Deep Learning?
It is one of the most popular software platforms used for deep learning and contains powerful tools to help you build and implement artificial neural networks.
Advancements in deep learning are being seen in smartphone applications, creating efficiencies in the power grid, driving advancements in healthcare, improving agricultural yields, and helping us find solutions to climate change. With this Tensorflow course, you’ll build expertise in deep learning models, learn to operate TensorFlow to manage neural networks and interpret the results.
And according to payscale.com, the median salary for engineers with deep learning skills tops $120,000 per year.
You can gain in-depth knowledge of Deep Learning by taking our Deep Learning certification training course. With Simplilearn’s Deep Learning course, you will prepare for a career as a Deep Learning engineer as you master concepts and techniques including supervised and unsupervised learning, mathematical and heuristic aspects, and hands-on modeling to develop algorithms. Those who complete the course will be able to:
Learn more at: https://www.simplilearn.com/
Machine learning and linear regression programmingSoumya Mukherjee
Overview of AI and ML
Terminology awareness
Applications in real world
Use cases within Nokia
Types of Learning
Regression
Classification
Clustering
Linear Regression Single Variable with python
Decision Tree Algorithm With Example | Decision Tree In Machine Learning | Da...Simplilearn
This Decision Tree Algorithm in Machine Learning Presentation will help you understand all the basics of Decision Tree along with what is Machine Learning, problems in Machine Learning, what is Decision Tree, advantages and disadvantages of Decision Tree, how Decision Tree algorithm works with solved examples and at the end we will implement a Decision Tree use case/ demo in Python on loan payment prediction. This Decision Tree tutorial is ideal for both beginners as well as professionals who want to learn Machine Learning Algorithms.
Below topics are covered in this Decision Tree Algorithm Presentation:
1. What is Machine Learning?
2. Types of Machine Learning?
3. Problems in Machine Learning
4. What is Decision Tree?
5. What are the problems a Decision Tree Solves?
6. Advantages of Decision Tree
7. How does Decision Tree Work?
8. Use Case - Loan Repayment Prediction
What is Machine Learning: Machine Learning is an application of Artificial Intelligence (AI) that provides systems the ability to automatically learn and improve from experience without being explicitly programmed.
- - - - - - - -
About Simplilearn Machine Learning course:
A form of artificial intelligence, Machine Learning is revolutionizing the world of computing as well as all people’s digital interactions. Machine Learning powers such innovative automated technologies as recommendation engines, facial recognition, fraud protection and even self-driving cars.This Machine Learning course prepares engineers, data scientists and other professionals with knowledge and hands-on skills required for certification and job competency in Machine Learning.
- - - - - - -
Why learn Machine Learning?
Machine Learning is taking over the world- and with that, there is a growing need among companies for professionals to know the ins and outs of Machine Learning
The Machine Learning market size is expected to grow from USD 1.03 Billion in 2016 to USD 8.81 Billion by 2022, at a Compound Annual Growth Rate (CAGR) of 44.1% during the forecast period.
- - - - - -
What skills will you learn from this Machine Learning course?
By the end of this Machine Learning course, you will be able to:
1. Master the concepts of supervised, unsupervised and reinforcement learning concepts and modeling.
2. Gain practical mastery over principles, algorithms, and applications of Machine Learning through a hands-on approach which includes working on 28 projects and one capstone project.
3. Acquire thorough knowledge of the mathematical and heuristic aspects of Machine Learning.
4. Understand the concepts and operation of support vector machines, kernel SVM, naive bayes, decision tree classifier, random forest classifier, logistic regression, K-nearest neighbors, K-means clustering and more.
5. Be able to model a wide variety of robust Machine Learning algorithms including deep learning, clustering, and recommendation systems
- - - - - - -
In machine learning, support vector machines (SVMs, also support-vector networks) are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis.
Introduction to Statistical Machine Learningmahutte
This course provides a broad introduction to the methods and practice
of statistical machine learning, which is concerned with the development
of algorithms and techniques that learn from observed data by
constructing stochastic models that can be used for making predictions
and decisions. Topics covered include Bayesian inference and maximum
likelihood modeling; regression, classi¯cation, density estimation,
clustering, principal component analysis; parametric, semi-parametric,
and non-parametric models; basis functions, neural networks, kernel
methods, and graphical models; deterministic and stochastic
optimization; over¯tting, regularization, and validation.
A Support Vector Machine (SVM) is a discriminative classifier formally defined by a separating hyperplane. In other words, given labeled training data (supervised learning), the algorithm outputs an optimal hyperplane which categorizes new examples. In two dimentional space this hyperplane is a line dividing a plane in two parts where in each class lay in either side.
Machine Learning with Apache Flink at Stockholm Machine Learning GroupTill Rohrmann
This presentation presents Apache Flink's approach to scalable machine learning: Composable machine learning pipelines, consisting of transformers and learners, and distributed linear algebra.
The presentation was held at the Machine Learning Stockholm group on the 23rd of March 2015.
Logistic Regression in Python | Logistic Regression Example | Machine Learnin...Edureka!
** Python Data Science Training : https://www.edureka.co/python **
This Edureka Video on Logistic Regression in Python will give you basic understanding of Logistic Regression Machine Learning Algorithm with examples. In this video, you will also get to see demo on Logistic Regression using Python. Below are the topics covered in this tutorial:
1. What is Regression?
2. What is Logistic Regression?
3. Why use Logistic Regression?
4. Linear vs Logistic Regression
5. Logistic Regression Use Cases
6. Logistic Regression Example Demo in Python
Subscribe to our channel to get video updates. Hit the subscribe button above.
Machine Learning Tutorial Playlist: https://goo.gl/UxjTxm
k-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster. This results in a partitioning of the data space into Voronoi cells.
Machine Learning Tutorial Part - 1 | Machine Learning Tutorial For Beginners ...Simplilearn
This presentation on Machine Learning will help you understand why Machine Learning came into picture, what is Machine Learning, types of Machine Learning, Machine Learning algorithms with a detailed explanation on linear regression, decision tree & support vector machine and at the end you will also see a use case implementation where we classify whether a recipe is of a cupcake or muffin using SVM algorithm. Machine learning is a core sub-area of artificial intelligence; it enables computers to get into a mode of self-learning without being explicitly programmed. When exposed to new data, these computer programs are enabled to learn, grow, change, and develop by themselves. So, to put simply, the iterative aspect of machine learning is the ability to adapt to new data independently. Now, let us get started with this Machine Learning presentation and understand what it is and why it matters.
Below topics are explained in this Machine Learning presentation:
1. Why Machine Learning?
2. What is Machine Learning?
3. Types of Machine Learning
4. Machine Learning Algorithms
- Linear Regression
- Decision Trees
- Support Vector Machine
5. Use case: Classify whether a recipe is of a cupcake or a muffin using SVM
About Simplilearn Machine Learning course:
A form of artificial intelligence, Machine Learning is revolutionizing the world of computing as well as all people’s digital interactions. Machine Learning powers such innovative automated technologies as recommendation engines, facial recognition, fraud protection and even self-driving cars. This Machine Learning course prepares engineers, data scientists and other professionals with knowledge and hands-on skills required for certification and job competency in Machine Learning.
Why learn Machine Learning?
Machine Learning is taking over the world- and with that, there is a growing need among companies for professionals to know the ins and outs of Machine Learning
The Machine Learning market size is expected to grow from USD 1.03 Billion in 2016 to USD 8.81 Billion by 2022, at a Compound Annual Growth Rate (CAGR) of 44.1% during the forecast period.
We recommend this Machine Learning training course for the following professionals in particular:
1. Developers aspiring to be a data scientist or Machine Learning engineer
2. Information architects who want to gain expertise in Machine Learning algorithms
3. Analytics professionals who want to work in Machine Learning or artificial intelligence
4. Graduates looking to build a career in data science and Machine Learning
Learn more at: https://www.simplilearn.com/
Linear Regression vs Logistic Regression | EdurekaEdureka!
YouTube: https://youtu.be/OCwZyYH14uw
** Data Science Certification using R: https://www.edureka.co/data-science **
This Edureka PPT on Linear Regression Vs Logistic Regression covers the basic concepts of linear and logistic models. The following topics are covered in this session:
Types of Machine Learning
Regression Vs Classification
What is Linear Regression?
What is Logistic Regression?
Linear Regression Use Case
Logistic Regression Use Case
Linear Regression Vs Logistic Regression
Blog Series: http://bit.ly/data-science-blogs
Data Science Training Playlist: http://bit.ly/data-science-playlist
Follow us to never miss an update in the future.
YouTube: https://www.youtube.com/user/edurekaIN
Instagram: https://www.instagram.com/edureka_learning/
Facebook: https://www.facebook.com/edurekaIN/
Twitter: https://twitter.com/edurekain
LinkedIn: https://www.linkedin.com/company/edureka
In machine learning, support-vector machines (SVMs, also support-vector networks) are supervised learning models with associated learning algorithms that analyze data for classification and regression analysis
Convolutional Neural Network - CNN | How CNN Works | Deep Learning Course | S...Simplilearn
This presentation on Convolutional neural network tutorial (CNN) will help you understand what is a convolutional neural network, hoe CNN recognizes images, what are layers in the convolutional neural network and at the end, you will see a use case implementation using CNN. CNN is a feed forward neural network that is generally used to analyze visual images by processing data with grid like topology. A CNN is also known as a "ConvNet". Convolutional networks can also perform optical character recognition to digitize text and make natural-language processing possible on analog and hand-written documents. CNNs can also be applied to sound when it is represented visually as a spectrogram. Now, lets deep dive into this presentation to understand what is CNN and how do they actually work.
Below topics are explained in this CNN presentation(Convolutional Neural Network presentation)
1. Introduction to CNN
2. What is a convolutional neural network?
3. How CNN recognizes images?
4. Layers in convolutional neural network
5. Use case implementation using CNN
Simplilearn’s Deep Learning course will transform you into an expert in deep learning techniques using TensorFlow, the open-source software library designed to conduct machine learning & deep neural network research. With our deep learning course, you’ll master deep learning and TensorFlow concepts, learn to implement algorithms, build artificial neural networks and traverse layers of data abstraction to understand the power of data and prepare you for your new role as deep learning scientist.
Why Deep Learning?
It is one of the most popular software platforms used for deep learning and contains powerful tools to help you build and implement artificial neural networks.
Advancements in deep learning are being seen in smartphone applications, creating efficiencies in the power grid, driving advancements in healthcare, improving agricultural yields, and helping us find solutions to climate change. With this Tensorflow course, you’ll build expertise in deep learning models, learn to operate TensorFlow to manage neural networks and interpret the results.
And according to payscale.com, the median salary for engineers with deep learning skills tops $120,000 per year.
You can gain in-depth knowledge of Deep Learning by taking our Deep Learning certification training course. With Simplilearn’s Deep Learning course, you will prepare for a career as a Deep Learning engineer as you master concepts and techniques including supervised and unsupervised learning, mathematical and heuristic aspects, and hands-on modeling to develop algorithms. Those who complete the course will be able to:
Learn more at: https://www.simplilearn.com/
Machine learning and linear regression programmingSoumya Mukherjee
Overview of AI and ML
Terminology awareness
Applications in real world
Use cases within Nokia
Types of Learning
Regression
Classification
Clustering
Linear Regression Single Variable with python
Decision Tree Algorithm With Example | Decision Tree In Machine Learning | Da...Simplilearn
This Decision Tree Algorithm in Machine Learning Presentation will help you understand all the basics of Decision Tree along with what is Machine Learning, problems in Machine Learning, what is Decision Tree, advantages and disadvantages of Decision Tree, how Decision Tree algorithm works with solved examples and at the end we will implement a Decision Tree use case/ demo in Python on loan payment prediction. This Decision Tree tutorial is ideal for both beginners as well as professionals who want to learn Machine Learning Algorithms.
Below topics are covered in this Decision Tree Algorithm Presentation:
1. What is Machine Learning?
2. Types of Machine Learning?
3. Problems in Machine Learning
4. What is Decision Tree?
5. What are the problems a Decision Tree Solves?
6. Advantages of Decision Tree
7. How does Decision Tree Work?
8. Use Case - Loan Repayment Prediction
What is Machine Learning: Machine Learning is an application of Artificial Intelligence (AI) that provides systems the ability to automatically learn and improve from experience without being explicitly programmed.
- - - - - - - -
About Simplilearn Machine Learning course:
A form of artificial intelligence, Machine Learning is revolutionizing the world of computing as well as all people’s digital interactions. Machine Learning powers such innovative automated technologies as recommendation engines, facial recognition, fraud protection and even self-driving cars.This Machine Learning course prepares engineers, data scientists and other professionals with knowledge and hands-on skills required for certification and job competency in Machine Learning.
- - - - - - -
Why learn Machine Learning?
Machine Learning is taking over the world- and with that, there is a growing need among companies for professionals to know the ins and outs of Machine Learning
The Machine Learning market size is expected to grow from USD 1.03 Billion in 2016 to USD 8.81 Billion by 2022, at a Compound Annual Growth Rate (CAGR) of 44.1% during the forecast period.
- - - - - -
What skills will you learn from this Machine Learning course?
By the end of this Machine Learning course, you will be able to:
1. Master the concepts of supervised, unsupervised and reinforcement learning concepts and modeling.
2. Gain practical mastery over principles, algorithms, and applications of Machine Learning through a hands-on approach which includes working on 28 projects and one capstone project.
3. Acquire thorough knowledge of the mathematical and heuristic aspects of Machine Learning.
4. Understand the concepts and operation of support vector machines, kernel SVM, naive bayes, decision tree classifier, random forest classifier, logistic regression, K-nearest neighbors, K-means clustering and more.
5. Be able to model a wide variety of robust Machine Learning algorithms including deep learning, clustering, and recommendation systems
- - - - - - -
In machine learning, support vector machines (SVMs, also support-vector networks) are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis.
Introduction to Statistical Machine Learningmahutte
This course provides a broad introduction to the methods and practice
of statistical machine learning, which is concerned with the development
of algorithms and techniques that learn from observed data by
constructing stochastic models that can be used for making predictions
and decisions. Topics covered include Bayesian inference and maximum
likelihood modeling; regression, classi¯cation, density estimation,
clustering, principal component analysis; parametric, semi-parametric,
and non-parametric models; basis functions, neural networks, kernel
methods, and graphical models; deterministic and stochastic
optimization; over¯tting, regularization, and validation.
A Support Vector Machine (SVM) is a discriminative classifier formally defined by a separating hyperplane. In other words, given labeled training data (supervised learning), the algorithm outputs an optimal hyperplane which categorizes new examples. In two dimentional space this hyperplane is a line dividing a plane in two parts where in each class lay in either side.
Machine Learning with Apache Flink at Stockholm Machine Learning GroupTill Rohrmann
This presentation presents Apache Flink's approach to scalable machine learning: Composable machine learning pipelines, consisting of transformers and learners, and distributed linear algebra.
The presentation was held at the Machine Learning Stockholm group on the 23rd of March 2015.
I am Fabian H. I am a Calculus Homework Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics, Deakin University, Australia. I have been helping students with their homework for the past 6 years. I solve homework related to Calculus.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Calculus Homework.
In this paper, natural inner product structure for the space of fuzzy n−tuples is introduced. Also we have
introduced the ortho vector, stochastic fuzzy vectors, ortho- stochastic fuzzy vectors, ortho-stochastic fuzzy
matrices and the concept of orthogonal complement of fuzzy vector subspace of a fuzzy vector space.
Math for Intelligent Systems - 01 Linear Algebra 01 Vector SpacesAndres Mendez-Vazquez
These are the initial notes for a class I am preparing for this summer in the Mathematics of Intelligent Systems. we will start with the vectors spaces, their basis and dimensions. The, we will look at one the basic applications the linear regression.
I am Manuela B. I am a Linear Algebra Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, the University of Warwick. I have been helping students with their assignments for the past 9 years. I solve assignments related to Linear Algebra.
Visit mathsassignmenthelp.com or email info@mathsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Linear Algebra Assignment.
Vector Space & Sub Space Presentation
Presented By: Sufian Mehmood Soomro
Department: (BS) Computer Science
Course Title: Linear Algebra
Shah Abdul Latif University Ghotki Campus
Senior data scientist and founder of the company Intelligentia Data I+D SA de CV. We are offering consultancy services, development of projects and products in Machine Learning, Big Data, Data Sciences and Artificial Intelligence.
My first set of slides (The NN and DL class I am preparing for the fall)... I included the problem of Vanishing Gradient and the need to have ReLu (Mentioning btw the saturation problem inherited from Hebbian Learning)
It has been almost 62 years since the invention of the term Artificial Intelligence by Samuel and Minsky et al. at the Dartmouth workshop College in 1956 (“Dartmouth Summer Research Project on Artificial Intelligence”) where this new area of Computer Science was invented. However, the history of Artificial Intelligence goes back to previous millennia, when the Greeks in their Myths spoke about golden robots at Hephaestus, and the Galatea of Pygmalion. They were the first automatons known at the dawn of history, and although these first attempts were only myths, automatons were invented and built through multiple civilizations in history. Nevertheless, these automatons resembled in quite limited way their final objectives, representing animals and humans. In spite of that, the greatest illusion of an automaton, the Turk by Wolfgang von Kempelen, inspired many people, trough its exhibitions, as Alexander Graham Bell and Charles Babbage to develop inventions that would change forever human history. Thus, the importance of the concept “Artificial Intelligence” as a driver of our technological dreams. And although Artificial Intelligence has never been defined in a precise practical way, the amount of research and methods that have been developed to tackle some of its basics tasks have been and are quite humongous. Thus, the importance of having an introduction to the concepts of Artificial Intelligence, thus the dream can continue.
A review of one of the most popular methods of clustering, a part of what is know as unsupervised learning, K-Means. Here, we go from the basic heuristic used to solve the NP-Hard problem to an approximation algorithm K-Centers. Additionally, we look at variations coming from the Fuzzy Set ideas. In the future, we will add more about On-Line algorithms in the line of Stochastic Gradient Ideas...
Here a Review of the Combination of Machine Learning models from Bayesian Averaging, Committees to Boosting... Specifically An statistical analysis of Boosting is done
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
1. Machine Learning for Data Mining
Linear Algebra Review
Andres Mendez-Vazquez
May 14, 2015
1 / 50
2. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
2 / 50
3. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
3 / 50
4. What is a Vector?
A ordered tuple of numbers
x =
x1
x2
...
xn
Expressing a magnitude and a direction
4 / 50
5. What is a Vector?
A ordered tuple of numbers
x =
x1
x2
...
xn
Expressing a magnitude and a direction
Magnitude
Direction
4 / 50
6. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
5 / 50
7. Vector Spaces
Definition
A vector is an element of a vector space
Vector Space V
It is a set that contains all linear combinations of its elements:
1 If x, y ∈ V then x + y ∈ V .
2 If x ∈ V then αx ∈ V for any scalar α.
3 There exists 0 ∈ V then x + 0 = x for any x ∈ V .
A subspace
It is a subset of a vector space that is also a vector space
6 / 50
8. Vector Spaces
Definition
A vector is an element of a vector space
Vector Space V
It is a set that contains all linear combinations of its elements:
1 If x, y ∈ V then x + y ∈ V .
2 If x ∈ V then αx ∈ V for any scalar α.
3 There exists 0 ∈ V then x + 0 = x for any x ∈ V .
A subspace
It is a subset of a vector space that is also a vector space
6 / 50
9. Vector Spaces
Definition
A vector is an element of a vector space
Vector Space V
It is a set that contains all linear combinations of its elements:
1 If x, y ∈ V then x + y ∈ V .
2 If x ∈ V then αx ∈ V for any scalar α.
3 There exists 0 ∈ V then x + 0 = x for any x ∈ V .
A subspace
It is a subset of a vector space that is also a vector space
6 / 50
10. Vector Spaces
Definition
A vector is an element of a vector space
Vector Space V
It is a set that contains all linear combinations of its elements:
1 If x, y ∈ V then x + y ∈ V .
2 If x ∈ V then αx ∈ V for any scalar α.
3 There exists 0 ∈ V then x + 0 = x for any x ∈ V .
A subspace
It is a subset of a vector space that is also a vector space
6 / 50
11. Vector Spaces
Definition
A vector is an element of a vector space
Vector Space V
It is a set that contains all linear combinations of its elements:
1 If x, y ∈ V then x + y ∈ V .
2 If x ∈ V then αx ∈ V for any scalar α.
3 There exists 0 ∈ V then x + 0 = x for any x ∈ V .
A subspace
It is a subset of a vector space that is also a vector space
6 / 50
13. Span
Definition
The span of any set of vectors {x1, x2, ..., xn} is defined as:
span (x1, x2, ..., xn) = α1x1 + α2x2 + ... + αnxn
What Examples can you Imagine?
Give it a shot!!!
8 / 50
14. Span
Definition
The span of any set of vectors {x1, x2, ..., xn} is defined as:
span (x1, x2, ..., xn) = α1x1 + α2x2 + ... + αnxn
What Examples can you Imagine?
Give it a shot!!!
8 / 50
15. Subspaces of Rn
A line through the origin in Rn
A plane in Rn
9 / 50
16. Subspaces of Rn
A line through the origin in Rn
A plane in Rn
9 / 50
17. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
10 / 50
18. Linear Independence and Basis of Vector Spaces
Fact 1
A vector x is a linearly independent of a set of vectors {x1, x2, ..., xn} if it
does not lie in their span.
Fact 2
A set of vectors is linearly independent if every vector is linearly
independent of the rest.
The Rest
1 A basis of a vector space V is a linearly independent set of vectors
whose span is equal to V
2 If the basis has d vectors then the vector space V has dimensionality
d.
11 / 50
19. Linear Independence and Basis of Vector Spaces
Fact 1
A vector x is a linearly independent of a set of vectors {x1, x2, ..., xn} if it
does not lie in their span.
Fact 2
A set of vectors is linearly independent if every vector is linearly
independent of the rest.
The Rest
1 A basis of a vector space V is a linearly independent set of vectors
whose span is equal to V
2 If the basis has d vectors then the vector space V has dimensionality
d.
11 / 50
20. Linear Independence and Basis of Vector Spaces
Fact 1
A vector x is a linearly independent of a set of vectors {x1, x2, ..., xn} if it
does not lie in their span.
Fact 2
A set of vectors is linearly independent if every vector is linearly
independent of the rest.
The Rest
1 A basis of a vector space V is a linearly independent set of vectors
whose span is equal to V
2 If the basis has d vectors then the vector space V has dimensionality
d.
11 / 50
21. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
12 / 50
22. Norm of a Vector
Definition
A norm u measures the magnitud of the vector.
Properties
1 Homogeneity: αx = α x .
2 Triangle inequality: x + y ≤ x + y .
3 Point Separation x = 0 if and only if x = 0.
Examples
1 Manhattan or 1-norm : x 1 = d
i=1 |xi|.
2 Euclidean or 2-norm : x 2 = d
i=1 x2
i .
13 / 50
23. Norm of a Vector
Definition
A norm u measures the magnitud of the vector.
Properties
1 Homogeneity: αx = α x .
2 Triangle inequality: x + y ≤ x + y .
3 Point Separation x = 0 if and only if x = 0.
Examples
1 Manhattan or 1-norm : x 1 = d
i=1 |xi|.
2 Euclidean or 2-norm : x 2 = d
i=1 x2
i .
13 / 50
24. Norm of a Vector
Definition
A norm u measures the magnitud of the vector.
Properties
1 Homogeneity: αx = α x .
2 Triangle inequality: x + y ≤ x + y .
3 Point Separation x = 0 if and only if x = 0.
Examples
1 Manhattan or 1-norm : x 1 = d
i=1 |xi|.
2 Euclidean or 2-norm : x 2 = d
i=1 x2
i .
13 / 50
25. Norm of a Vector
Definition
A norm u measures the magnitud of the vector.
Properties
1 Homogeneity: αx = α x .
2 Triangle inequality: x + y ≤ x + y .
3 Point Separation x = 0 if and only if x = 0.
Examples
1 Manhattan or 1-norm : x 1 = d
i=1 |xi|.
2 Euclidean or 2-norm : x 2 = d
i=1 x2
i .
13 / 50
26. Norm of a Vector
Definition
A norm u measures the magnitud of the vector.
Properties
1 Homogeneity: αx = α x .
2 Triangle inequality: x + y ≤ x + y .
3 Point Separation x = 0 if and only if x = 0.
Examples
1 Manhattan or 1-norm : x 1 = d
i=1 |xi|.
2 Euclidean or 2-norm : x 2 = d
i=1 x2
i .
13 / 50
27. Norm of a Vector
Definition
A norm u measures the magnitud of the vector.
Properties
1 Homogeneity: αx = α x .
2 Triangle inequality: x + y ≤ x + y .
3 Point Separation x = 0 if and only if x = 0.
Examples
1 Manhattan or 1-norm : x 1 = d
i=1 |xi|.
2 Euclidean or 2-norm : x 2 = d
i=1 x2
i .
13 / 50
29. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
15 / 50
30. Inner Product
Definition
The inner product between u and v
u, v =
n
i=1
uivi.
It is the projection of one vector onto the other one
Remark: It is related to the Euclidean norm: u, u = u 2
2.
16 / 50
31. Inner Product
Definition
The inner product between u and v
u, v =
n
i=1
uivi.
It is the projection of one vector onto the other one
16 / 50
32. Properties
Meaning
The inner product is a measure of correlation between two vectors, scaled
by the norms of the vectors
if u · v > 0, u and v are aligned
17 / 50
33. Properties
Meaning
The inner product is a measure of correlation between two vectors, scaled
by the norms of the vectors
if u · v > 0, u and v are aligned
17 / 50
34. Properties
The inner product is a measure of correlation between two vectors,
scaled by the norms of the vectors
18 / 50
35. Properties
The inner product is a measure of correlation between two vectors,
scaled by the norms of the vectors
19 / 50
36. Properties
The inner product is a measure of correlation between two vectors,
scaled by the norms of the vectors
20 / 50
37. Definitions involving the norm
Orthonormal
The vectors in orthonormal basis have unit Euclidean norm and are
orthonorgonal.
To express a vector x in an orthonormal basis
For example, given x = α1b1 + α2b2
x, b1 = α1b1 + α2b2, b1
= α1 b1, b1 + α2 b2, b1
= α1 + 0
Likewise, x, b2 = α2
21 / 50
38. Definitions involving the norm
Orthonormal
The vectors in orthonormal basis have unit Euclidean norm and are
orthonorgonal.
To express a vector x in an orthonormal basis
For example, given x = α1b1 + α2b2
x, b1 = α1b1 + α2b2, b1
= α1 b1, b1 + α2 b2, b1
= α1 + 0
Likewise, x, b2 = α2
21 / 50
39. Definitions involving the norm
Orthonormal
The vectors in orthonormal basis have unit Euclidean norm and are
orthonorgonal.
To express a vector x in an orthonormal basis
For example, given x = α1b1 + α2b2
x, b1 = α1b1 + α2b2, b1
= α1 b1, b1 + α2 b2, b1
= α1 + 0
Likewise, x, b2 = α2
21 / 50
40. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
22 / 50
41. Linear Operator
Definition
A linear operator L : U → V is a map from a vector space U to another
vector space V satisfies:
L (u1 + u2) = L (u1) + L (u2)
Something Notable
If the dimension n of U and m of V are finite, L can be represented by
m × n matrix:
A =
a11 a12 · · · a1n
a21 a22 · · · a2n
· · ·
am1 am2 · · · amn
23 / 50
42. Linear Operator
Definition
A linear operator L : U → V is a map from a vector space U to another
vector space V satisfies:
L (u1 + u2) = L (u1) + L (u2)
Something Notable
If the dimension n of U and m of V are finite, L can be represented by
m × n matrix:
A =
a11 a12 · · · a1n
a21 a22 · · · a2n
· · ·
am1 am2 · · · amn
23 / 50
43. Thus, product of
The product of two linear operator can be seen as the multiplication
of two matrices
AB =
a11 a12 · · · a1n
a21 a22 · · · a2n
· · ·
am1 am2 · · · amn
b11 b12 · · · b1p
b21 b22 · · · b2p
· · ·
bn1 bn2 · · · bnp
=
n
i=1 a1ibi1
n
i=1 a1ibi2 · · · n
i=1 a1ibip
n
i=1 a2ibi1
n
i=1 a2ibi2 · · · n
i=1 a2ibip
· · ·
n
i=1 amibi1
n
i=1 amibi2 · · · n
i=1 amibip
Note: if A is m × n and B is n × p, then AB is m × p.
24 / 50
44. Thus, product of
The product of two linear operator can be seen as the multiplication
of two matrices
AB =
a11 a12 · · · a1n
a21 a22 · · · a2n
· · ·
am1 am2 · · · amn
b11 b12 · · · b1p
b21 b22 · · · b2p
· · ·
bn1 bn2 · · · bnp
=
n
i=1 a1ibi1
n
i=1 a1ibi2 · · · n
i=1 a1ibip
n
i=1 a2ibi1
n
i=1 a2ibi2 · · · n
i=1 a2ibip
· · ·
n
i=1 amibi1
n
i=1 amibi2 · · · n
i=1 amibip
Note: if A is m × n and B is n × p, then AB is m × p.
24 / 50
45. Transpose of a Matrix
The transpose of a matrix is obtained by flipping the rows and
columns
AT
=
a11 a21 · · · an1
a12 a22 · · · an2
· · ·
a1m a2m · · · anm
Which the following properties
AT
T
= A
(A + B)T
= AT + BT
(AB)T
= BT AT
Not only that, we have the inner product
u, v = uT
v
25 / 50
46. Transpose of a Matrix
The transpose of a matrix is obtained by flipping the rows and
columns
AT
=
a11 a21 · · · an1
a12 a22 · · · an2
· · ·
a1m a2m · · · anm
Which the following properties
AT
T
= A
(A + B)T
= AT + BT
(AB)T
= BT AT
Not only that, we have the inner product
u, v = uT
v
25 / 50
47. Transpose of a Matrix
The transpose of a matrix is obtained by flipping the rows and
columns
AT
=
a11 a21 · · · an1
a12 a22 · · · an2
· · ·
a1m a2m · · · anm
Which the following properties
AT
T
= A
(A + B)T
= AT + BT
(AB)T
= BT AT
Not only that, we have the inner product
u, v = uT
v
25 / 50
48. Transpose of a Matrix
The transpose of a matrix is obtained by flipping the rows and
columns
AT
=
a11 a21 · · · an1
a12 a22 · · · an2
· · ·
a1m a2m · · · anm
Which the following properties
AT
T
= A
(A + B)T
= AT + BT
(AB)T
= BT AT
Not only that, we have the inner product
u, v = uT
v
25 / 50
49. Transpose of a Matrix
The transpose of a matrix is obtained by flipping the rows and
columns
AT
=
a11 a21 · · · an1
a12 a22 · · · an2
· · ·
a1m a2m · · · anm
Which the following properties
AT
T
= A
(A + B)T
= AT + BT
(AB)T
= BT AT
Not only that, we have the inner product
u, v = uT
v
25 / 50
50. As always, we have the identity operator
The identity operator in matrix multiplication is defined as
I =
1 0 · · · 0
0 1 · · · 0
· · ·
0 0 · · · 1
With properties
For any matrix A, AI = A.
I is the identity operator for the matrix product.
26 / 50
51. As always, we have the identity operator
The identity operator in matrix multiplication is defined as
I =
1 0 · · · 0
0 1 · · · 0
· · ·
0 0 · · · 1
With properties
For any matrix A, AI = A.
I is the identity operator for the matrix product.
26 / 50
52. As always, we have the identity operator
The identity operator in matrix multiplication is defined as
I =
1 0 · · · 0
0 1 · · · 0
· · ·
0 0 · · · 1
With properties
For any matrix A, AI = A.
I is the identity operator for the matrix product.
26 / 50
53. Column Space, Row Space and Rank
Let A be an m × n matrix
We have the following spaces...
Column space
Span of the columns of A.
Linear subspace of Rm.
Row space
Span of the rows of A.
Linear subspace of Rn.
27 / 50
54. Column Space, Row Space and Rank
Let A be an m × n matrix
We have the following spaces...
Column space
Span of the columns of A.
Linear subspace of Rm.
Row space
Span of the rows of A.
Linear subspace of Rn.
27 / 50
55. Column Space, Row Space and Rank
Let A be an m × n matrix
We have the following spaces...
Column space
Span of the columns of A.
Linear subspace of Rm.
Row space
Span of the rows of A.
Linear subspace of Rn.
27 / 50
56. Column Space, Row Space and Rank
Let A be an m × n matrix
We have the following spaces...
Column space
Span of the columns of A.
Linear subspace of Rm.
Row space
Span of the rows of A.
Linear subspace of Rn.
27 / 50
57. Important facts
Something Notable
The column and row space of any matrix have the same dimension.
The rank
The dimension is the rank of the matrix.
28 / 50
58. Important facts
Something Notable
The column and row space of any matrix have the same dimension.
The rank
The dimension is the rank of the matrix.
28 / 50
59. Range and Null Space
Range
Set of vectors equal to Au for some u ∈ Rn.
Range (A) = {x|x = Au for some u ∈ Rn
}
It is a linear subspace of Rm and also called the column space of A.
Null Space
We have the following definition
Null Space (A) = {u|Au = 0}
It is a linear subspace of Rm.
29 / 50
60. Range and Null Space
Range
Set of vectors equal to Au for some u ∈ Rn.
Range (A) = {x|x = Au for some u ∈ Rn
}
It is a linear subspace of Rm and also called the column space of A.
Null Space
We have the following definition
Null Space (A) = {u|Au = 0}
It is a linear subspace of Rm.
29 / 50
61. Range and Null Space
Range
Set of vectors equal to Au for some u ∈ Rn.
Range (A) = {x|x = Au for some u ∈ Rn
}
It is a linear subspace of Rm and also called the column space of A.
Null Space
We have the following definition
Null Space (A) = {u|Au = 0}
It is a linear subspace of Rm.
29 / 50
62. Range and Null Space
Range
Set of vectors equal to Au for some u ∈ Rn.
Range (A) = {x|x = Au for some u ∈ Rn
}
It is a linear subspace of Rm and also called the column space of A.
Null Space
We have the following definition
Null Space (A) = {u|Au = 0}
It is a linear subspace of Rm.
29 / 50
63. Important fact
Something Notable
Every vector in the null space is orthogonal to the rows of A.
The null space and row space of a matrix are orthogonal.
30 / 50
64. Important fact
Something Notable
Every vector in the null space is orthogonal to the rows of A.
The null space and row space of a matrix are orthogonal.
30 / 50
65. Range and Column Space
We have another interpretation of the matrix-vector product
Au = (A1 A2 · · · An)
u1
u2
...
un
=u1A1 + u2A2 + · · · + unAn
Thus
The result is a linear combination of the columns of A.
Actually, the range is the column space.
31 / 50
66. Range and Column Space
We have another interpretation of the matrix-vector product
Au = (A1 A2 · · · An)
u1
u2
...
un
=u1A1 + u2A2 + · · · + unAn
Thus
The result is a linear combination of the columns of A.
Actually, the range is the column space.
31 / 50
67. Range and Column Space
We have another interpretation of the matrix-vector product
Au = (A1 A2 · · · An)
u1
u2
...
un
=u1A1 + u2A2 + · · · + unAn
Thus
The result is a linear combination of the columns of A.
Actually, the range is the column space.
31 / 50
68. Matrix Inverse
Something Notable
For an n × n matrix A: rank + dim(null space) = n.
if dim(null space)= 0 then A is full rank.
In this case, the action of the matrix is invertible.
The inversion is also linear and consequently can be represented by
another matrix A−1.
A−1 is the only matrix such that A−1A = AA−1 = I.
32 / 50
69. Matrix Inverse
Something Notable
For an n × n matrix A: rank + dim(null space) = n.
if dim(null space)= 0 then A is full rank.
In this case, the action of the matrix is invertible.
The inversion is also linear and consequently can be represented by
another matrix A−1.
A−1 is the only matrix such that A−1A = AA−1 = I.
32 / 50
70. Matrix Inverse
Something Notable
For an n × n matrix A: rank + dim(null space) = n.
if dim(null space)= 0 then A is full rank.
In this case, the action of the matrix is invertible.
The inversion is also linear and consequently can be represented by
another matrix A−1.
A−1 is the only matrix such that A−1A = AA−1 = I.
32 / 50
71. Matrix Inverse
Something Notable
For an n × n matrix A: rank + dim(null space) = n.
if dim(null space)= 0 then A is full rank.
In this case, the action of the matrix is invertible.
The inversion is also linear and consequently can be represented by
another matrix A−1.
A−1 is the only matrix such that A−1A = AA−1 = I.
32 / 50
72. Matrix Inverse
Something Notable
For an n × n matrix A: rank + dim(null space) = n.
if dim(null space)= 0 then A is full rank.
In this case, the action of the matrix is invertible.
The inversion is also linear and consequently can be represented by
another matrix A−1.
A−1 is the only matrix such that A−1A = AA−1 = I.
32 / 50
73. Orthogonal Matrices
Definition
An orthogonal matrix U satisfies UT U = I.
Properties
U has orthonormal columns.
In addition
Applying an orthogonal matrix to two vectors does not change their inner
product:
Uu, Uv = (Uu)T
Uv
=uT
UT
Uv
=uT
v
= u, v
33 / 50
74. Orthogonal Matrices
Definition
An orthogonal matrix U satisfies UT U = I.
Properties
U has orthonormal columns.
In addition
Applying an orthogonal matrix to two vectors does not change their inner
product:
Uu, Uv = (Uu)T
Uv
=uT
UT
Uv
=uT
v
= u, v
33 / 50
75. Orthogonal Matrices
Definition
An orthogonal matrix U satisfies UT U = I.
Properties
U has orthonormal columns.
In addition
Applying an orthogonal matrix to two vectors does not change their inner
product:
Uu, Uv = (Uu)T
Uv
=uT
UT
Uv
=uT
v
= u, v
33 / 50
77. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
35 / 50
78. Trace and Determinant
Definition (Trace)
The trace is the sum of the diagonal elements of a square matrix.
Definition (Determinant)
The determinant of a square matrix A, denoted by |A|, is defined as
det (A) =
n
j=1
(−1)i+j
aijMij
where Mij is determinant of matrix A without the row i and column j.
36 / 50
79. Trace and Determinant
Definition (Trace)
The trace is the sum of the diagonal elements of a square matrix.
Definition (Determinant)
The determinant of a square matrix A, denoted by |A|, is defined as
det (A) =
n
j=1
(−1)i+j
aijMij
where Mij is determinant of matrix A without the row i and column j.
36 / 50
80. Special Case
For a 2 × 2 matrix A =
a b
c d
|A| = ad − bc
The absolute value of |A|is the area of the parallelogram given by the
rows of A
37 / 50
81. Special Case
For a 2 × 2 matrix A =
a b
c d
|A| = ad − bc
The absolute value of |A|is the area of the parallelogram given by the
rows of A
37 / 50
82. Properties of the Determinant
Basic Properties
|A| = AT
|AB| = |A| |B|
|A| = 0 if and only if A is not invertible
If A is invertible, then A−1 = 1
|A| .
38 / 50
83. Properties of the Determinant
Basic Properties
|A| = AT
|AB| = |A| |B|
|A| = 0 if and only if A is not invertible
If A is invertible, then A−1 = 1
|A| .
38 / 50
84. Properties of the Determinant
Basic Properties
|A| = AT
|AB| = |A| |B|
|A| = 0 if and only if A is not invertible
If A is invertible, then A−1 = 1
|A| .
38 / 50
85. Properties of the Determinant
Basic Properties
|A| = AT
|AB| = |A| |B|
|A| = 0 if and only if A is not invertible
If A is invertible, then A−1 = 1
|A| .
38 / 50
86. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
39 / 50
87. Eigenvalues and Eigenvectors
Eigenvalues
An eigenvalue λ of a square matrix A satisfies:
Au = λu
for some vector , which we call an eigenvector.
Properties
Geometrically the operator A expands when (λ > 1) or contracts (λ < 1)
eigenvectors, but does not rotate them.
Null Space relation
If u is an eigenvector of A, it is in the null space of A − λI, which is
consequently not invertible.
40 / 50
88. Eigenvalues and Eigenvectors
Eigenvalues
An eigenvalue λ of a square matrix A satisfies:
Au = λu
for some vector , which we call an eigenvector.
Properties
Geometrically the operator A expands when (λ > 1) or contracts (λ < 1)
eigenvectors, but does not rotate them.
Null Space relation
If u is an eigenvector of A, it is in the null space of A − λI, which is
consequently not invertible.
40 / 50
89. Eigenvalues and Eigenvectors
Eigenvalues
An eigenvalue λ of a square matrix A satisfies:
Au = λu
for some vector , which we call an eigenvector.
Properties
Geometrically the operator A expands when (λ > 1) or contracts (λ < 1)
eigenvectors, but does not rotate them.
Null Space relation
If u is an eigenvector of A, it is in the null space of A − λI, which is
consequently not invertible.
40 / 50
90. More properties
Given the previous relation
The eigenvalues of A are the roots of the equation |A − λI| = 0
Remark: We do not calculate the eigenvalues this way
Something Notable
Eigenvalues and eigenvectors can be complex valued, even if all the entries
of A are real.
41 / 50
91. More properties
Given the previous relation
The eigenvalues of A are the roots of the equation |A − λI| = 0
Remark: We do not calculate the eigenvalues this way
Something Notable
Eigenvalues and eigenvectors can be complex valued, even if all the entries
of A are real.
41 / 50
92. Eigendecomposition of a Matrix
Given
Let A be an n × n square matrix with n linearly independent eigenvectors
p1, p2, ..., pn and eigenvalues λ1, λ2, ..., λn
We define the matrices
P = (p1 p2 · · · pn)
Λ =
λ1 0 · · · 0
0 λ2 · · · 0
· · ·
0 0 · · · λn
42 / 50
93. Eigendecomposition of a Matrix
Given
Let A be an n × n square matrix with n linearly independent eigenvectors
p1, p2, ..., pn and eigenvalues λ1, λ2, ..., λn
We define the matrices
P = (p1 p2 · · · pn)
Λ =
λ1 0 · · · 0
0 λ2 · · · 0
· · ·
0 0 · · · λn
42 / 50
94. Properties
We have that A satisfies
AP = PΛ
In addition
P is full rank.
Thus, inverting it yields the eigendecomposition
A = PΛP−1
43 / 50
95. Properties
We have that A satisfies
AP = PΛ
In addition
P is full rank.
Thus, inverting it yields the eigendecomposition
A = PΛP−1
43 / 50
96. Properties
We have that A satisfies
AP = PΛ
In addition
P is full rank.
Thus, inverting it yields the eigendecomposition
A = PΛP−1
43 / 50
97. Properties of the Eigendecomposition
We have that
Not all matrices are diagonalizable/eigendecomposition. Example
1 1
0 1
Trace (A) = Trace (Λ) = n
i=1 λi
|A| = |Λ| = Πn
i=1λi
The rank of A is equal to the number of nonzero eigenvalues.
If λ is anonzero eigenvalue of A, 1
λ is an eigenvalue of A−1 with the
same eigenvector.
The eigendecompositon allows to compute matrix powers efficiently:
Am = PΛP−1 m
= PΛP−1PΛP−1PΛP−1 . . . PΛP−1 =
PΛmP−1
44 / 50
98. Properties of the Eigendecomposition
We have that
Not all matrices are diagonalizable/eigendecomposition. Example
1 1
0 1
Trace (A) = Trace (Λ) = n
i=1 λi
|A| = |Λ| = Πn
i=1λi
The rank of A is equal to the number of nonzero eigenvalues.
If λ is anonzero eigenvalue of A, 1
λ is an eigenvalue of A−1 with the
same eigenvector.
The eigendecompositon allows to compute matrix powers efficiently:
Am = PΛP−1 m
= PΛP−1PΛP−1PΛP−1 . . . PΛP−1 =
PΛmP−1
44 / 50
99. Properties of the Eigendecomposition
We have that
Not all matrices are diagonalizable/eigendecomposition. Example
1 1
0 1
Trace (A) = Trace (Λ) = n
i=1 λi
|A| = |Λ| = Πn
i=1λi
The rank of A is equal to the number of nonzero eigenvalues.
If λ is anonzero eigenvalue of A, 1
λ is an eigenvalue of A−1 with the
same eigenvector.
The eigendecompositon allows to compute matrix powers efficiently:
Am = PΛP−1 m
= PΛP−1PΛP−1PΛP−1 . . . PΛP−1 =
PΛmP−1
44 / 50
100. Properties of the Eigendecomposition
We have that
Not all matrices are diagonalizable/eigendecomposition. Example
1 1
0 1
Trace (A) = Trace (Λ) = n
i=1 λi
|A| = |Λ| = Πn
i=1λi
The rank of A is equal to the number of nonzero eigenvalues.
If λ is anonzero eigenvalue of A, 1
λ is an eigenvalue of A−1 with the
same eigenvector.
The eigendecompositon allows to compute matrix powers efficiently:
Am = PΛP−1 m
= PΛP−1PΛP−1PΛP−1 . . . PΛP−1 =
PΛmP−1
44 / 50
101. Properties of the Eigendecomposition
We have that
Not all matrices are diagonalizable/eigendecomposition. Example
1 1
0 1
Trace (A) = Trace (Λ) = n
i=1 λi
|A| = |Λ| = Πn
i=1λi
The rank of A is equal to the number of nonzero eigenvalues.
If λ is anonzero eigenvalue of A, 1
λ is an eigenvalue of A−1 with the
same eigenvector.
The eigendecompositon allows to compute matrix powers efficiently:
Am = PΛP−1 m
= PΛP−1PΛP−1PΛP−1 . . . PΛP−1 =
PΛmP−1
44 / 50
102. Properties of the Eigendecomposition
We have that
Not all matrices are diagonalizable/eigendecomposition. Example
1 1
0 1
Trace (A) = Trace (Λ) = n
i=1 λi
|A| = |Λ| = Πn
i=1λi
The rank of A is equal to the number of nonzero eigenvalues.
If λ is anonzero eigenvalue of A, 1
λ is an eigenvalue of A−1 with the
same eigenvector.
The eigendecompositon allows to compute matrix powers efficiently:
Am = PΛP−1 m
= PΛP−1PΛP−1PΛP−1 . . . PΛP−1 =
PΛmP−1
44 / 50
103. Eigendecomposition of a Symmetric Matrix
When A symmetric, we have
If A = AT then A is symmetric.
The eigenvalues of symmetric matrices are real.
The eigenvectors of symmetric matrices are orthonormal.
Consequently, the eigendecomposition becomes A = UΛUT for Λ
real and U orthogonal.
The eigenvectors of A are an orthonormal basis for the column space
and row space.
45 / 50
104. Eigendecomposition of a Symmetric Matrix
When A symmetric, we have
If A = AT then A is symmetric.
The eigenvalues of symmetric matrices are real.
The eigenvectors of symmetric matrices are orthonormal.
Consequently, the eigendecomposition becomes A = UΛUT for Λ
real and U orthogonal.
The eigenvectors of A are an orthonormal basis for the column space
and row space.
45 / 50
105. Eigendecomposition of a Symmetric Matrix
When A symmetric, we have
If A = AT then A is symmetric.
The eigenvalues of symmetric matrices are real.
The eigenvectors of symmetric matrices are orthonormal.
Consequently, the eigendecomposition becomes A = UΛUT for Λ
real and U orthogonal.
The eigenvectors of A are an orthonormal basis for the column space
and row space.
45 / 50
106. Eigendecomposition of a Symmetric Matrix
When A symmetric, we have
If A = AT then A is symmetric.
The eigenvalues of symmetric matrices are real.
The eigenvectors of symmetric matrices are orthonormal.
Consequently, the eigendecomposition becomes A = UΛUT for Λ
real and U orthogonal.
The eigenvectors of A are an orthonormal basis for the column space
and row space.
45 / 50
107. Eigendecomposition of a Symmetric Matrix
When A symmetric, we have
If A = AT then A is symmetric.
The eigenvalues of symmetric matrices are real.
The eigenvectors of symmetric matrices are orthonormal.
Consequently, the eigendecomposition becomes A = UΛUT for Λ
real and U orthogonal.
The eigenvectors of A are an orthonormal basis for the column space
and row space.
45 / 50
108. We can see the action of a symmetric matrix on a vector u
as...
We can decompose the action Au = UΛUT
u as
Projection of u onto the column space of A (Multiplication by UT ).
Scaling of each coefficient Ui, u by the corresponding eigenvalue
(Multiplication by Λ).
Linear combination of the eigenvectors scaled by the resulting
coefficient (Multiplication by U).
Final equation
Au =
n
i=1
λi Ui, u Ui
It would be great to generalize this to all matrices!!!
46 / 50
109. We can see the action of a symmetric matrix on a vector u
as...
We can decompose the action Au = UΛUT
u as
Projection of u onto the column space of A (Multiplication by UT ).
Scaling of each coefficient Ui, u by the corresponding eigenvalue
(Multiplication by Λ).
Linear combination of the eigenvectors scaled by the resulting
coefficient (Multiplication by U).
Final equation
Au =
n
i=1
λi Ui, u Ui
It would be great to generalize this to all matrices!!!
46 / 50
110. We can see the action of a symmetric matrix on a vector u
as...
We can decompose the action Au = UΛUT
u as
Projection of u onto the column space of A (Multiplication by UT ).
Scaling of each coefficient Ui, u by the corresponding eigenvalue
(Multiplication by Λ).
Linear combination of the eigenvectors scaled by the resulting
coefficient (Multiplication by U).
Final equation
Au =
n
i=1
λi Ui, u Ui
It would be great to generalize this to all matrices!!!
46 / 50
111. We can see the action of a symmetric matrix on a vector u
as...
We can decompose the action Au = UΛUT
u as
Projection of u onto the column space of A (Multiplication by UT ).
Scaling of each coefficient Ui, u by the corresponding eigenvalue
(Multiplication by Λ).
Linear combination of the eigenvectors scaled by the resulting
coefficient (Multiplication by U).
Final equation
Au =
n
i=1
λi Ui, u Ui
It would be great to generalize this to all matrices!!!
46 / 50
112. We can see the action of a symmetric matrix on a vector u
as...
We can decompose the action Au = UΛUT
u as
Projection of u onto the column space of A (Multiplication by UT ).
Scaling of each coefficient Ui, u by the corresponding eigenvalue
(Multiplication by Λ).
Linear combination of the eigenvectors scaled by the resulting
coefficient (Multiplication by U).
Final equation
Au =
n
i=1
λi Ui, u Ui
It would be great to generalize this to all matrices!!!
46 / 50
113. Outline
1 Introduction
What is a Vector?
2 Vector Spaces
Definition
Linear Independence and Basis of Vector Spaces
Norm of a Vector
Inner Product
Matrices
Trace and Determinant
Matrix Decomposition
Singular Value Decomposition
47 / 50
114. Singular Value Decomposition
Every Matrix has a singular value decomposition
A = UΣV T
Where
The columns of U are an orthonormal basis for the column space.
The columns of V are an orthonormal basis for the row space.
The Σ is diagonal and the entries on its diagonal σi = Σii are positive
real numbers, called the singular values of A.
The action of Aon a vector u can be decomposed into
Au = n
i=1 σi Vi, u Ui
48 / 50
115. Singular Value Decomposition
Every Matrix has a singular value decomposition
A = UΣV T
Where
The columns of U are an orthonormal basis for the column space.
The columns of V are an orthonormal basis for the row space.
The Σ is diagonal and the entries on its diagonal σi = Σii are positive
real numbers, called the singular values of A.
The action of Aon a vector u can be decomposed into
Au = n
i=1 σi Vi, u Ui
48 / 50
116. Singular Value Decomposition
Every Matrix has a singular value decomposition
A = UΣV T
Where
The columns of U are an orthonormal basis for the column space.
The columns of V are an orthonormal basis for the row space.
The Σ is diagonal and the entries on its diagonal σi = Σii are positive
real numbers, called the singular values of A.
The action of Aon a vector u can be decomposed into
Au = n
i=1 σi Vi, u Ui
48 / 50
117. Singular Value Decomposition
Every Matrix has a singular value decomposition
A = UΣV T
Where
The columns of U are an orthonormal basis for the column space.
The columns of V are an orthonormal basis for the row space.
The Σ is diagonal and the entries on its diagonal σi = Σii are positive
real numbers, called the singular values of A.
The action of Aon a vector u can be decomposed into
Au = n
i=1 σi Vi, u Ui
48 / 50
118. Properties of the Singular Value Decomposition
First
The eigenvalues of the symmetric matrix AT A are equal to the square of
the singular values of A:
AT A = V ΣUT UT ΣV T = V Σ2V T
Second
The rank of a matrix is equal to the number of nonzero singular values.
Third
The largest singular value σ1 is the solution to the optimization problem:
σ1 = max
x=0
Ax 2
x 2
49 / 50
119. Properties of the Singular Value Decomposition
First
The eigenvalues of the symmetric matrix AT A are equal to the square of
the singular values of A:
AT A = V ΣUT UT ΣV T = V Σ2V T
Second
The rank of a matrix is equal to the number of nonzero singular values.
Third
The largest singular value σ1 is the solution to the optimization problem:
σ1 = max
x=0
Ax 2
x 2
49 / 50
120. Properties of the Singular Value Decomposition
First
The eigenvalues of the symmetric matrix AT A are equal to the square of
the singular values of A:
AT A = V ΣUT UT ΣV T = V Σ2V T
Second
The rank of a matrix is equal to the number of nonzero singular values.
Third
The largest singular value σ1 is the solution to the optimization problem:
σ1 = max
x=0
Ax 2
x 2
49 / 50
121. Properties of the Singular Value Decomposition
Remark
It can be verified that the largest singular value satisfies the properties of a
norm, it is called the spectral norm of the matrix.
Finally
In statistics analyzing data with the singular value decomposition is called
Principal Component Analysis.
50 / 50
122. Properties of the Singular Value Decomposition
Remark
It can be verified that the largest singular value satisfies the properties of a
norm, it is called the spectral norm of the matrix.
Finally
In statistics analyzing data with the singular value decomposition is called
Principal Component Analysis.
50 / 50