This document provides an introduction to systems of linear equations and matrix operations. It defines key concepts such as matrices, matrix addition and multiplication, and transitions between different bases. It presents an example of multiplying two matrices using NumPy. The document outlines how systems of linear equations can be represented using matrices and discusses solving systems using techniques like Gauss-Jordan elimination and elementary row operations. It also introduces the concepts of homogeneous and inhomogeneous systems.
Here is a basic Linear Algebra review for the class of Machine Learning. This is actually becoming a new class in the mathematics of Intelligent Systems, there I will be teaching stuff in
1.- Linear Algebra - From the basics to the Cayley-Hamilton Theorem with applications
2.- Mathematical Analysis - from set to the Reimann Integral
3.- Topology - Mostly in Hilbert Spaces
4.- Optimization - Convex functions, KKT conditions, Duality Theory, etc.
The stuff is going to be interesting...
Here is a basic Linear Algebra review for the class of Machine Learning. This is actually becoming a new class in the mathematics of Intelligent Systems, there I will be teaching stuff in
1.- Linear Algebra - From the basics to the Cayley-Hamilton Theorem with applications
2.- Mathematical Analysis - from set to the Reimann Integral
3.- Topology - Mostly in Hilbert Spaces
4.- Optimization - Convex functions, KKT conditions, Duality Theory, etc.
The stuff is going to be interesting...
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni 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 fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
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
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
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
I am Anthony F. I am a Digital Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, University of Cambridge, UK. I have been helping students with their homework for the past 8 years. I solve assignments related to Digital Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Digital Signal Processing Assignments.
Eigen values and eigen vectors engineeringshubham211
mathematics...for engineering mathematics.....learn maths...............................The individual items in a matrix are called its elements or entries.[4] Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element. The rule for matrix multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field. A major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation which can be represented by a rotation matrix R: if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two transformation matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a linear transformation is obtainable (along with other information) from the matrix's eigenvalues and eigenvectors.
Applications of matrices are found in most scientific fields. In every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, they are used to study physical phenomena, such as the motion of rigid bodies. In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.[5] Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. Matrix decomposition methods simplify computations, both theoretically and practically. Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function
...
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni 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 fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
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
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
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
I am Anthony F. I am a Digital Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, University of Cambridge, UK. I have been helping students with their homework for the past 8 years. I solve assignments related to Digital Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Digital Signal Processing Assignments.
Linear Algebra may be defined as the form of algebra in which there is a study of different kinds of solutions which are related to linear equations. In order to explain the Linear Algebra, it is important to explain that the title consists of two different terms. The very first term which is important to be considered in the same, is Linear. Linear may be defined as something which is straight. Linear equations can be used for the calculation of the equation in a xy plane where the straight lines has been defined. In addition to this, linear equations can be used to define something which is straight in a three dimensional perspective. Another view of linear equations may be defined as flatness which recognizes the set of points which can be used for giving the description related to the equations which are in a very simple forms. These are the equations which involves the addition and multiplication.
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
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
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.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
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.
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.
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. Mathematics for Artificial Intelligence
System of Linear Equations
Andres Mendez-Vazquez
March 2, 2020
1 / 96
2. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
2 / 96
3. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
3 / 96
4. As we saw in the previous introduction
The Development of Linear Algebra
It is as a natural extension of trying to solve systems of linear equations.
From those early attempts - Gauss and Company
Cayley have the need to formalize fully the concept of Matrices,
From this simple concept a new era in Mathematics would arise.
4 / 96
5. As we saw in the previous introduction
The Development of Linear Algebra
It is as a natural extension of trying to solve systems of linear equations.
From those early attempts - Gauss and Company
Cayley have the need to formalize fully the concept of Matrices,
From this simple concept a new era in Mathematics would arise.
4 / 96
6. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
5 / 96
7. Example
A classic problem is to solve systems of linear equations like
3x + 3y = 12
x − 2y = 1
With
6 / 96
8. Example
A classic problem is to solve systems of linear equations like
3x + 3y = 12
x − 2y = 1
With
0
1 2 3
1
2
4
6 / 96
9. However
It is clear that once the dimension of the vector space increases
beyond two
We do not have a simple geometric method to solve this problem.
We can use our knowledge of Matrices
3 3
1 −2
x
y
=
12
1
Thus, we have the equation in the following format Ax = y
x =
x
y
, y =
12
1
and A =
3 3
1 −2
7 / 96
10. However
It is clear that once the dimension of the vector space increases
beyond two
We do not have a simple geometric method to solve this problem.
We can use our knowledge of Matrices
3 3
1 −2
x
y
=
12
1
Thus, we have the equation in the following format Ax = y
x =
x
y
, y =
12
1
and A =
3 3
1 −2
7 / 96
11. However
It is clear that once the dimension of the vector space increases
beyond two
We do not have a simple geometric method to solve this problem.
We can use our knowledge of Matrices
3 3
1 −2
x
y
=
12
1
Thus, we have the equation in the following format Ax = y
x =
x
y
, y =
12
1
and A =
3 3
1 −2
7 / 96
12. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
8 / 96
13. Definition of Matrices
Definition
Let K be a field, and let n, m be two integers≥ 1. An array of scalars in
K:
a11 a12 · · · a1n
a21 a22 · · · a2n
...
...
...
...
am1 am2 · · · amn
is called a matrix in K. We can abbreviate the notation writing (aij) ,
i = 1, ..., m and j = 1, ..., n.
9 / 96
14. Further
We call aij the ij−entry of the matrix, and the ith
row is defined as
Ai = (ai1, ai2, ..., ain)
The jth
column is denoted as
Aj
=
a1j
a2j
...
amj
10 / 96
15. Further
We call aij the ij−entry of the matrix, and the ith
row is defined as
Ai = (ai1, ai2, ..., ain)
The jth
column is denoted as
Aj
=
a1j
a2j
...
amj
10 / 96
16. Addition of Matrices
Definition
Let A = (aij) and B = (bij) be two m × n matrices. We define A + B be
a matrix whose entry in the ith row and jth column is aij + bij.
Therefore, Is this possible?
1 −1 0
2 3 4
+
5 1
2 1
=?
11 / 96
17. Addition of Matrices
Definition
Let A = (aij) and B = (bij) be two m × n matrices. We define A + B be
a matrix whose entry in the ith row and jth column is aij + bij.
Therefore, Is this possible?
1 −1 0
2 3 4
+
5 1
2 1
=?
11 / 96
18. The Zero Matrix
Definition
Let A = (aij) be q m × n matrix whose entries are all 0. This matrix is
the zero matrix, 0mn.
Example
Look at the Jupyter...
12 / 96
19. The Zero Matrix
Definition
Let A = (aij) be q m × n matrix whose entries are all 0. This matrix is
the zero matrix, 0mn.
Example
Look at the Jupyter...
12 / 96
20. Multiplication By Scalar
Definition
The multiplication by an scalar element which is defined simply as, given a
matrix A and scalar c, a matrix cA whose ij−component is caij.
Example at Jupyter
Remarks
It is easy to see that the set of matrices of size m × n with components in
a field K form a vector space over K which can be denoted by
Matm×n (K).
13 / 96
21. Multiplication By Scalar
Definition
The multiplication by an scalar element which is defined simply as, given a
matrix A and scalar c, a matrix cA whose ij−component is caij.
Example at Jupyter
Remarks
It is easy to see that the set of matrices of size m × n with components in
a field K form a vector space over K which can be denoted by
Matm×n (K).
13 / 96
22. Some Other Definitions
Definition
Let A = (aij) be an m × n matrix. The matrix B = (bij) such that
bji = aijis called transpose of A, and is also denoted by AT .
Example at Jupyter
Additionally
A matrix is said to be symmetric if it is equal to its transpose i.e. if
AT = A.
14 / 96
23. Some Other Definitions
Definition
Let A = (aij) be an m × n matrix. The matrix B = (bij) such that
bji = aijis called transpose of A, and is also denoted by AT .
Example at Jupyter
Additionally
A matrix is said to be symmetric if it is equal to its transpose i.e. if
AT = A.
14 / 96
24. Matrix Multiplication
Definition
If the number of columns of A (m × k) equals the number of rows of B
(k × n), then the product C = AB is defined by
cij =
k
h=1
aikbkj (1)
15 / 96
27. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
18 / 96
28. A Change on Basis
We can do the following
You have a vector x in certain space V with a basis B = {v1, v2, ..., vn}.
Thus, we have
x = a1v1 + a2v2 + · · · + anvn
19 / 96
29. A Change on Basis
We can do the following
You have a vector x in certain space V with a basis B = {v1, v2, ..., vn}.
Thus, we have
x = a1v1 + a2v2 + · · · + anvn
19 / 96
30. We have
Then in the basis B = {v1, v2, ..., vn}
[x]B =
a1
a2
...
an
B
20 / 96
31. We need to solve the following system of equations
Solving the following system
a1v1 + a2v2 + · · · + anvn = x
Or
v1 v2 · · · vn
a1
a2
...
an
B
= xst
21 / 96
32. We need to solve the following system of equations
Solving the following system
a1v1 + a2v2 + · · · + anvn = x
Or
v1 v2 · · · vn
a1
a2
...
an
B
= xst
21 / 96
33. What if...?
if we have another basis for such space A = {w1, w2, ..., wn}
w1 = b11v1 + b21v2 + · · · + bn1vm
w2 = b12v1 + b22v2 + · · · + bn2vm
... =
...
wn = b1nv1 + b2nw2 + · · · + bnnvm
Therefore, we generate the following matrix
b11 b12 · · · b1n
b21 b22 · · · b2n
...
...
...
...
bn1 bn2 · · · bnn
22 / 96
34. What if...?
if we have another basis for such space A = {w1, w2, ..., wn}
w1 = b11v1 + b21v2 + · · · + bn1vm
w2 = b12v1 + b22v2 + · · · + bn2vm
... =
...
wn = b1nv1 + b2nw2 + · · · + bnnvm
Therefore, we generate the following matrix
b11 b12 · · · b1n
b21 b22 · · · b2n
...
...
...
...
bn1 bn2 · · · bnn
22 / 96
35. Therefore
We have that each column represent a vector wj in standard basis
wj =
b1j
b2j
...
bnj
23 / 96
38. With the following property
Then using the B = {v1, v2, ..., vn} representation
ciwi = cib1iv1 + cib2iv2 + · · · + cibnivm
Therefore
(c1b11 + ... + cnb1n) v1 + · · · + (c1bn1 + ... + cnbnn) vn = [x]B
25 / 96
39. With the following property
Then using the B = {v1, v2, ..., vn} representation
ciwi = cib1iv1 + cib2iv2 + · · · + cibnivm
Therefore
(c1b11 + ... + cnb1n) v1 + · · · + (c1bn1 + ... + cnbnn) vn = [x]B
25 / 96
40. Or
The Coordinates for the System in basis {v1, v2, ..., vn}
Nice, What about {v1, v2, ..., vn} → {w1, w2, ..., wn}?
But Normally, we want to go from {v1, v2, ..., vn} to
{w1, w2, ..., wn}
Simply, given d1 d2 · · · dn
T
c1
c2
...
cn
=
b11 b12 · · · b1n
b21 b22 · · · b2n
...
...
...
...
bn1 bn2 · · · bnn
−1
d1
d2
...
dn
26 / 96
41. Or
The Coordinates for the System in basis {v1, v2, ..., vn}
Nice, What about {v1, v2, ..., vn} → {w1, w2, ..., wn}?
But Normally, we want to go from {v1, v2, ..., vn} to
{w1, w2, ..., wn}
Simply, given d1 d2 · · · dn
T
c1
c2
...
cn
=
b11 b12 · · · b1n
b21 b22 · · · b2n
...
...
...
...
bn1 bn2 · · · bnn
−1
d1
d2
...
dn
26 / 96
42. Or
The Coordinates for the System in basis {v1, v2, ..., vn}
Nice, What about {v1, v2, ..., vn} → {w1, w2, ..., wn}?
But Normally, we want to go from {v1, v2, ..., vn} to
{w1, w2, ..., wn}
Simply, given d1 d2 · · · dn
T
c1
c2
...
cn
=
b11 b12 · · · b1n
b21 b22 · · · b2n
...
...
...
...
bn1 bn2 · · · bnn
−1
d1
d2
...
dn
26 / 96
43. Then
This is basically a way to represent the change of basis
By inner product, multiplication of matrices, inverses and matrix-vector
multiplication.
27 / 96
44. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
28 / 96
47. Therefore, we can see that
We can calculate the determinant, det = 0 linear independence
det
1 2 3
2 −1 2
−1 4 1
= 0
We derive the standard coordinates of v
v =
1 2 3
2 −1 2
−1 4 1
4
1
−5
B
30 / 96
48. Therefore, we can see that
We can calculate the determinant, det = 0 linear independence
det
1 2 3
2 −1 2
−1 4 1
= 0
We derive the standard coordinates of v
v =
1 2 3
2 −1 2
−1 4 1
4
1
−5
B
30 / 96
49. Now, we have that to move from one basis to another
What if we have a element in basis S
[v]S =
5
7
−3
We derive the B coordinates of vector v
5
7
−3
==
1 2 3
2 −1 2
−1 4 1
a1
a2
a3
31 / 96
50. Now, we have that to move from one basis to another
What if we have a element in basis S
[v]S =
5
7
−3
We derive the B coordinates of vector v
5
7
−3
==
1 2 3
2 −1 2
−1 4 1
a1
a2
a3
31 / 96
51. Then, we have
Solve the system or get the inverse
1 2 3
2 −1 2
−1 4 1
−1
5
7
−3
=
a1
a2
a3
Make the following example
B =
1
0
0
,
0
1
0
,
0
0
1
32 / 96
52. Then, we have
Solve the system or get the inverse
1 2 3
2 −1 2
−1 4 1
−1
5
7
−3
=
a1
a2
a3
Make the following example
B =
1
0
0
,
0
1
0
,
0
0
1
32 / 96
53. Change of basis from B to B
Given an old basis B of Rn
with transition matrix PB
And a new basis B with transition matrix PB
How do we change from coords in the basis B to coords in the basis
B ?
Coordinates in B, then using v = PB [v]B we change the coordinates
to standard coordinates.
Then, we can do
[v]B = P−1
B v
33 / 96
54. Change of basis from B to B
Given an old basis B of Rn
with transition matrix PB
And a new basis B with transition matrix PB
How do we change from coords in the basis B to coords in the basis
B ?
Coordinates in B, then using v = PB [v]B we change the coordinates
to standard coordinates.
Then, we can do
[v]B = P−1
B v
33 / 96
55. Change of basis from B to B
Given an old basis B of Rn
with transition matrix PB
And a new basis B with transition matrix PB
How do we change from coords in the basis B to coords in the basis
B ?
Coordinates in B, then using v = PB [v]B we change the coordinates
to standard coordinates.
Then, we can do
[v]B = P−1
B v
33 / 96
56. Therefore, we have
We have the following situation
[v]B = P−1
B PB [v]B
Then, the final transition matrix
M = P−1
B PB = P−1
B v1 v2 · · · vn
In other words
M = P−1
B P−1
B v1 P−1
B v2 · · · P−1
B vn
34 / 96
57. Therefore, we have
We have the following situation
[v]B = P−1
B PB [v]B
Then, the final transition matrix
M = P−1
B PB = P−1
B v1 v2 · · · vn
In other words
M = P−1
B P−1
B v1 P−1
B v2 · · · P−1
B vn
34 / 96
58. Therefore, we have
We have the following situation
[v]B = P−1
B PB [v]B
Then, the final transition matrix
M = P−1
B PB = P−1
B v1 v2 · · · vn
In other words
M = P−1
B P−1
B v1 P−1
B v2 · · · P−1
B vn
34 / 96
59. Therefore, we have
Something Notable
The columns of the transition matrix M from the old basis B to the
new basis B
They are the coordinate vectors of the old basis B with respect to the
new basis B .
35 / 96
60. Fianlly, packing everything
Theorem
If B and B are two bases of Rn , with B = {v1, v2, · · · , vn} then
the transition matrix from B coordinates to B coordinates is given by
M = [[v1]B , [v2]B , ..., [vn]B ]
36 / 96
61. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
37 / 96
62. Going Back to the Problem
We have
3x + 3y
x − 2y
=
12
1
Nevertheless
This not simplify the way in which we solve it.
The variables x and y can be eliminated from the computation
By simply writing down a matrix in which:
1 The coefficients of x are in the first column.
2 The coefficients of y are in the second column.
3 The right hand side of the system is the third column.
38 / 96
63. Going Back to the Problem
We have
3x + 3y
x − 2y
=
12
1
Nevertheless
This not simplify the way in which we solve it.
The variables x and y can be eliminated from the computation
By simply writing down a matrix in which:
1 The coefficients of x are in the first column.
2 The coefficients of y are in the second column.
3 The right hand side of the system is the third column.
38 / 96
64. Going Back to the Problem
We have
3x + 3y
x − 2y
=
12
1
Nevertheless
This not simplify the way in which we solve it.
The variables x and y can be eliminated from the computation
By simply writing down a matrix in which:
1 The coefficients of x are in the first column.
2 The coefficients of y are in the second column.
3 The right hand side of the system is the third column.
38 / 96
65. Going Back to the Problem
We have
3x + 3y
x − 2y
=
12
1
Nevertheless
This not simplify the way in which we solve it.
The variables x and y can be eliminated from the computation
By simply writing down a matrix in which:
1 The coefficients of x are in the first column.
2 The coefficients of y are in the second column.
3 The right hand side of the system is the third column.
38 / 96
66. Going Back to the Problem
We have
3x + 3y
x − 2y
=
12
1
Nevertheless
This not simplify the way in which we solve it.
The variables x and y can be eliminated from the computation
By simply writing down a matrix in which:
1 The coefficients of x are in the first column.
2 The coefficients of y are in the second column.
3 The right hand side of the system is the third column.
38 / 96
67. Going Back to the Problem
We have
3x + 3y
x − 2y
=
12
1
Nevertheless
This not simplify the way in which we solve it.
The variables x and y can be eliminated from the computation
By simply writing down a matrix in which:
1 The coefficients of x are in the first column.
2 The coefficients of y are in the second column.
3 The right hand side of the system is the third column.
38 / 96
68. Therefore
We have (Basically columns as place makers)
3 3 12
1 −2 1
Then, look at the following
3 3 12
3 −6 3
: Multiply the second row by 3
Further
6 6 24
3 −6 3
: Multiply the first row by 2
39 / 96
69. Therefore
We have (Basically columns as place makers)
3 3 12
1 −2 1
Then, look at the following
3 3 12
3 −6 3
: Multiply the second row by 3
Further
6 6 24
3 −6 3
: Multiply the first row by 2
39 / 96
70. Therefore
We have (Basically columns as place makers)
3 3 12
1 −2 1
Then, look at the following
3 3 12
3 −6 3
: Multiply the second row by 3
Further
6 6 24
3 −6 3
: Multiply the first row by 2
39 / 96
71. Add row one and two
We have
6 6 24
9 0 27
Therefore x = 3
From it we can get y = 1.
Not Only that
All “equivalent” systems of equations have the same solutions.
40 / 96
72. Add row one and two
We have
6 6 24
9 0 27
Therefore x = 3
From it we can get y = 1.
Not Only that
All “equivalent” systems of equations have the same solutions.
40 / 96
73. Add row one and two
We have
6 6 24
9 0 27
Therefore x = 3
From it we can get y = 1.
Not Only that
All “equivalent” systems of equations have the same solutions.
40 / 96
74. Augmented Matrix
Definition
The previous matrix is called the augmented matrix of the system, and can
be written in matrix shorthand as (A|y).
Note
The previous system can be solved in different ways.
Additionally, the system of equations (Two Equations)
1 There’s just one,
2 There is no solution,
3 There are an infinite number of solutions.
41 / 96
75. Augmented Matrix
Definition
The previous matrix is called the augmented matrix of the system, and can
be written in matrix shorthand as (A|y).
Note
The previous system can be solved in different ways.
Additionally, the system of equations (Two Equations)
1 There’s just one,
2 There is no solution,
3 There are an infinite number of solutions.
41 / 96
76. Augmented Matrix
Definition
The previous matrix is called the augmented matrix of the system, and can
be written in matrix shorthand as (A|y).
Note
The previous system can be solved in different ways.
Additionally, the system of equations (Two Equations)
1 There’s just one,
2 There is no solution,
3 There are an infinite number of solutions.
41 / 96
77. Augmented Matrix
Definition
The previous matrix is called the augmented matrix of the system, and can
be written in matrix shorthand as (A|y).
Note
The previous system can be solved in different ways.
Additionally, the system of equations (Two Equations)
1 There’s just one,
2 There is no solution,
3 There are an infinite number of solutions.
41 / 96
78. Augmented Matrix
Definition
The previous matrix is called the augmented matrix of the system, and can
be written in matrix shorthand as (A|y).
Note
The previous system can be solved in different ways.
Additionally, the system of equations (Two Equations)
1 There’s just one,
2 There is no solution,
3 There are an infinite number of solutions.
41 / 96
79. What about?
Example
2x − 4y + z = 1
4x + y − z = 3
Then the augmented matrix
2 −4 1 1
4 1 −1 3
42 / 96
80. What about?
Example
2x − 4y + z = 1
4x + y − z = 3
Then the augmented matrix
2 −4 1 1
4 1 −1 3
42 / 96
81. Therefore
We have
1 −2 1
2
1
2
4 1 −1 3
: Mult row 1 by
1
2
Then
1 −2 1
2
1
2
0 9 −3 1
: Mult row 2 by -4 and add it to eqn 2
Further, we get a augmented matrix called an echelon form
1 −2 1
2
1
2
0 1 −1
3 1
: Mult row 2 by
1
9
43 / 96
82. Therefore
We have
1 −2 1
2
1
2
4 1 −1 3
: Mult row 1 by
1
2
Then
1 −2 1
2
1
2
0 9 −3 1
: Mult row 2 by -4 and add it to eqn 2
Further, we get a augmented matrix called an echelon form
1 −2 1
2
1
2
0 1 −1
3 1
: Mult row 2 by
1
9
43 / 96
83. Therefore
We have
1 −2 1
2
1
2
4 1 −1 3
: Mult row 1 by
1
2
Then
1 −2 1
2
1
2
0 9 −3 1
: Mult row 2 by -4 and add it to eqn 2
Further, we get a augmented matrix called an echelon form
1 −2 1
2
1
2
0 1 −1
3 1
: Mult row 2 by
1
9
43 / 96
84. Finally
We get a reduced echelon form
1 0 −1
6
13
18
0 1 −1
3 1
: Multiply Row 2 and add to row 1
We have clearly a free variable
It is the variable z
44 / 96
85. Finally
We get a reduced echelon form
1 0 −1
6
13
18
0 1 −1
3 1
: Multiply Row 2 and add to row 1
We have clearly a free variable
It is the variable z
44 / 96
86. What operations we have been doing in the augmented
matrices?
First
1 Multiply any equation by a non-zero real number (scalar).
2 Equivalent to multiplying a row of the matrix by a scalar.
Second
1 Replace any equation by the original equation plus a scalar multiple of
another equation.
2 Equivalent to replace any row of a matrix by that row plus a multiple
of another row.
Third
1 Interchange two equations.
2 Equivalent to two rows of the augmented matrix.
45 / 96
87. What operations we have been doing in the augmented
matrices?
First
1 Multiply any equation by a non-zero real number (scalar).
2 Equivalent to multiplying a row of the matrix by a scalar.
Second
1 Replace any equation by the original equation plus a scalar multiple of
another equation.
2 Equivalent to replace any row of a matrix by that row plus a multiple
of another row.
Third
1 Interchange two equations.
2 Equivalent to two rows of the augmented matrix.
45 / 96
88. What operations we have been doing in the augmented
matrices?
First
1 Multiply any equation by a non-zero real number (scalar).
2 Equivalent to multiplying a row of the matrix by a scalar.
Second
1 Replace any equation by the original equation plus a scalar multiple of
another equation.
2 Equivalent to replace any row of a matrix by that row plus a multiple
of another row.
Third
1 Interchange two equations.
2 Equivalent to two rows of the augmented matrix.
45 / 96
90. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
47 / 96
91. How we use only matrix operations?
Take a look at this example
A =
3 4 5
2 −1 0
We have the following elementary matrix coming from the identity
matrix
E1 =
1
3 0
0 1
We get
E1A =
1 4
3
5
3
2 −1 0
48 / 96
92. How we use only matrix operations?
Take a look at this example
A =
3 4 5
2 −1 0
We have the following elementary matrix coming from the identity
matrix
E1 =
1
3 0
0 1
We get
E1A =
1 4
3
5
3
2 −1 0
48 / 96
93. How we use only matrix operations?
Take a look at this example
A =
3 4 5
2 −1 0
We have the following elementary matrix coming from the identity
matrix
E1 =
1
3 0
0 1
We get
E1A =
1 4
3
5
3
2 −1 0
48 / 96
94. Further
To add −2×(row one) to row 2 in the identity matrix
E2 =
1 0
−2 1
Then
E2E1A =
1 4
3
5
3
0 −11
3 −10
3
Finally we can use the matrix E3 =
1 0
0 − 3
11
E3E2E1A =
1 4
3
5
3
0 1 10
11
49 / 96
95. Further
To add −2×(row one) to row 2 in the identity matrix
E2 =
1 0
−2 1
Then
E2E1A =
1 4
3
5
3
0 −11
3 −10
3
Finally we can use the matrix E3 =
1 0
0 − 3
11
E3E2E1A =
1 4
3
5
3
0 1 10
11
49 / 96
96. Further
To add −2×(row one) to row 2 in the identity matrix
E2 =
1 0
−2 1
Then
E2E1A =
1 4
3
5
3
0 −11
3 −10
3
Finally we can use the matrix E3 =
1 0
0 − 3
11
E3E2E1A =
1 4
3
5
3
0 1 10
11
49 / 96
97. Finally, we have
E4 =
1 −4
3
0 1
E4E3E2E1A =
1 0 5
11
0 1 10
11
Definition
The matrix R is said to be in echelon form provided that:
1 The leading entry of every non-zero row is a 1.
2 If the leading entry of row i is in position k, and the next row is not a
row of zeros, then the leading entry of row i + 1 is in position k + j,
where j ≥ 1.
3 All zero rows are at the bottom of the matrix.
50 / 96
98. Finally, we have
E4 =
1 −4
3
0 1
E4E3E2E1A =
1 0 5
11
0 1 10
11
Definition
The matrix R is said to be in echelon form provided that:
1 The leading entry of every non-zero row is a 1.
2 If the leading entry of row i is in position k, and the next row is not a
row of zeros, then the leading entry of row i + 1 is in position k + j,
where j ≥ 1.
3 All zero rows are at the bottom of the matrix.
50 / 96
99. Finally, we have
E4 =
1 −4
3
0 1
E4E3E2E1A =
1 0 5
11
0 1 10
11
Definition
The matrix R is said to be in echelon form provided that:
1 The leading entry of every non-zero row is a 1.
2 If the leading entry of row i is in position k, and the next row is not a
row of zeros, then the leading entry of row i + 1 is in position k + j,
where j ≥ 1.
3 All zero rows are at the bottom of the matrix.
50 / 96
100. Finally, we have
E4 =
1 −4
3
0 1
E4E3E2E1A =
1 0 5
11
0 1 10
11
Definition
The matrix R is said to be in echelon form provided that:
1 The leading entry of every non-zero row is a 1.
2 If the leading entry of row i is in position k, and the next row is not a
row of zeros, then the leading entry of row i + 1 is in position k + j,
where j ≥ 1.
3 All zero rows are at the bottom of the matrix.
50 / 96
101. Finally, we have
E4 =
1 −4
3
0 1
E4E3E2E1A =
1 0 5
11
0 1 10
11
Definition
The matrix R is said to be in echelon form provided that:
1 The leading entry of every non-zero row is a 1.
2 If the leading entry of row i is in position k, and the next row is not a
row of zeros, then the leading entry of row i + 1 is in position k + j,
where j ≥ 1.
3 All zero rows are at the bottom of the matrix.
50 / 96
102. Then
Examples
1 ∗
0 1
,
1 ∗ ∗
0 0 1
0 0 0
and
0 1 ∗ ∗
0 0 1 ∗
0 0 0 1
This leads to
R is in reduced echelon form (Gauss-Jordan Form) if
1 R is in echelon form
2 Each leading entry is the only non-zero entry in its column.
51 / 96
103. Then
Examples
1 ∗
0 1
,
1 ∗ ∗
0 0 1
0 0 0
and
0 1 ∗ ∗
0 0 1 ∗
0 0 0 1
This leads to
R is in reduced echelon form (Gauss-Jordan Form) if
1 R is in echelon form
2 Each leading entry is the only non-zero entry in its column.
51 / 96
104. Example
We have
1 0
0 1
,
1 ∗ 0 ∗
0 0 1 ∗
0 0 0 0
and
0 1 0 0 ∗
0 0 1 0 ∗
0 0 0 1 ∗
Question
Suppose A is n × m matrix. What is the maximum number of leading 1’s
that can appear when it’s been reduced to echelon form?
52 / 96
105. Example
We have
1 0
0 1
,
1 ∗ 0 ∗
0 0 1 ∗
0 0 0 0
and
0 1 0 0 ∗
0 0 1 0 ∗
0 0 0 1 ∗
Question
Suppose A is n × m matrix. What is the maximum number of leading 1’s
that can appear when it’s been reduced to echelon form?
52 / 96
106. From these
You can reduce a square matrix A into a Gauss-Jordan Form
Ek · · · E2E1A = I
What is the name of
B = Ek · · · E2E1
53 / 96
107. From these
You can reduce a square matrix A into a Gauss-Jordan Form
Ek · · · E2E1A = I
What is the name of
B = Ek · · · E2E1
53 / 96
108. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
54 / 96
109. Elementary Matrices
Definition
Elementary Matrices are constructed by performing the given row
operation on the identity matrix:
1 To multiply row j of A by the scalar c use the matrix E obtained
from I by multiplying jth row of I by c.
2 To add c × rowj (A) to rowk (A), use the identity matrix with its kth
row replaced by (0, ..., 0, c, 0, ..., 0, 1, 0, ...).
3 To interchange rows j and k, use the identity matrix with rows j and
k interchanged.
55 / 96
110. Elementary Matrices
Definition
Elementary Matrices are constructed by performing the given row
operation on the identity matrix:
1 To multiply row j of A by the scalar c use the matrix E obtained
from I by multiplying jth row of I by c.
2 To add c × rowj (A) to rowk (A), use the identity matrix with its kth
row replaced by (0, ..., 0, c, 0, ..., 0, 1, 0, ...).
3 To interchange rows j and k, use the identity matrix with rows j and
k interchanged.
55 / 96
111. Elementary Matrices
Definition
Elementary Matrices are constructed by performing the given row
operation on the identity matrix:
1 To multiply row j of A by the scalar c use the matrix E obtained
from I by multiplying jth row of I by c.
2 To add c × rowj (A) to rowk (A), use the identity matrix with its kth
row replaced by (0, ..., 0, c, 0, ..., 0, 1, 0, ...).
3 To interchange rows j and k, use the identity matrix with rows j and
k interchanged.
55 / 96
112. Elementary Matrices
Definition
Elementary Matrices are constructed by performing the given row
operation on the identity matrix:
1 To multiply row j of A by the scalar c use the matrix E obtained
from I by multiplying jth row of I by c.
2 To add c × rowj (A) to rowk (A), use the identity matrix with its kth
row replaced by (0, ..., 0, c, 0, ..., 0, 1, 0, ...).
3 To interchange rows j and k, use the identity matrix with rows j and
k interchanged.
55 / 96
113. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
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114. Properties
Elementary matrices are always square
Directly from the definition.
Elementary matrices are invertible
Basically you can revert the operations using another elementary matrix.
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115. Properties
Elementary matrices are always square
Directly from the definition.
Elementary matrices are invertible
Basically you can revert the operations using another elementary matrix.
57 / 96
116. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
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117. Therefore
Theorem
Elementary row operations applied to either Ax = y or the corresponding
augmented matrix (A|y) do not change the set of solutions to the system.
Proof
Given the augmented matrix (A|y), we multiply the by an elementary
matrix E
We get
E (A|y) = (EA|Ey)
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118. Therefore
Theorem
Elementary row operations applied to either Ax = y or the corresponding
augmented matrix (A|y) do not change the set of solutions to the system.
Proof
Given the augmented matrix (A|y), we multiply the by an elementary
matrix E
We get
E (A|y) = (EA|Ey)
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119. Therefore
Theorem
Elementary row operations applied to either Ax = y or the corresponding
augmented matrix (A|y) do not change the set of solutions to the system.
Proof
Given the augmented matrix (A|y), we multiply the by an elementary
matrix E
We get
E (A|y) = (EA|Ey)
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120. Then
We have this correspond to
EAx = Ey
Now, assume that x is a solution to Ax = y
Then, it solves the previous system!!!
Conversely
If x solves the new system, EAx = Ey, multiplication by E−1 gives
Ax = y
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121. Then
We have this correspond to
EAx = Ey
Now, assume that x is a solution to Ax = y
Then, it solves the previous system!!!
Conversely
If x solves the new system, EAx = Ey, multiplication by E−1 gives
Ax = y
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122. Then
We have this correspond to
EAx = Ey
Now, assume that x is a solution to Ax = y
Then, it solves the previous system!!!
Conversely
If x solves the new system, EAx = Ey, multiplication by E−1 gives
Ax = y
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123. Therefore
The end result of all the row operations on (A|y)
(EkEk−1 · · · E2E1A|EkEk−1 · · · E2E1y) = R
Where
R is an echelon form of (A|y).
Remark
And if R is in echelon form, we can easily work out the solution.
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124. Therefore
The end result of all the row operations on (A|y)
(EkEk−1 · · · E2E1A|EkEk−1 · · · E2E1y) = R
Where
R is an echelon form of (A|y).
Remark
And if R is in echelon form, we can easily work out the solution.
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125. Therefore
The end result of all the row operations on (A|y)
(EkEk−1 · · · E2E1A|EkEk−1 · · · E2E1y) = R
Where
R is an echelon form of (A|y).
Remark
And if R is in echelon form, we can easily work out the solution.
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126. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
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127. Algorithm
Gauss-Jordan (A, y)
1 Write the augmented matrix of the system.
2 Use row operations to transform the augmented matrix in the form
described below, which is called the reduced row echelon form:
1 The rows (if any) consisting entirely of zeros are grouped together at
the bottom of the matrix.
2 In each row that does not consist entirely of zeros, the leftmost
nonzero element is a 1 (called a leading 1 or a pivot).
3 Each column that contains a leading 1 has zeros in all other entries.
4 The leading 1 in any row is to the left of any leading 1’s in the rows
below it.
3 Stop process in step 2 if you obtain a row whose elements are all
zeros except the last one on the right. In that case, the system is
inconsistent and has no solutions.
4 Otherwise, finish step 2 and read the solutions of the system from the
final matrix.
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128. Example
Look at the Board for an example
Why not to try programming it!!!
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129. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
65 / 96
130. Some Definitions
Definition
A system of equations Ax = y is consistent if there is at least one
solution x.
If there is no solution, then the system is inconsistent.
Now Assume that the augmented (A|y) has been reduced
To either echelon or Gauss-Jordan Form
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131. Some Definitions
Definition
A system of equations Ax = y is consistent if there is at least one
solution x.
If there is no solution, then the system is inconsistent.
Now Assume that the augmented (A|y) has been reduced
To either echelon or Gauss-Jordan Form
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132. Gauss-Jordan Form?
Definition
The matrix A is upper triangular if any entry aij with i > j satisfies
aij = 0.
Thus
The row echelon form of the matrix is upper triangular.
Therefore
To continue the reduction to Gauss-Jordan form, it is only necessary to use
each leading 1 to clean out any remaining non-zero entries in its column.
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133. Gauss-Jordan Form?
Definition
The matrix A is upper triangular if any entry aij with i > j satisfies
aij = 0.
Thus
The row echelon form of the matrix is upper triangular.
Therefore
To continue the reduction to Gauss-Jordan form, it is only necessary to use
each leading 1 to clean out any remaining non-zero entries in its column.
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134. Gauss-Jordan Form?
Definition
The matrix A is upper triangular if any entry aij with i > j satisfies
aij = 0.
Thus
The row echelon form of the matrix is upper triangular.
Therefore
To continue the reduction to Gauss-Jordan form, it is only necessary to use
each leading 1 to clean out any remaining non-zero entries in its column.
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135. Example
Therefore
It is only necessary to use each leading 1 to clean out any remaining
non-zero entries in its column.
Example
1 ∗ 0 0 ∗
0 1 0 ∗
1 ∗
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136. Example
Therefore
It is only necessary to use each leading 1 to clean out any remaining
non-zero entries in its column.
Example
1 ∗ 0 0 ∗
0 1 0 ∗
1 ∗
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137. More Definitions
Definition
Suppose the augmented matrix for the linear system Ax = y has been
brought to echelon form.
If there is a leading 1 in any column except the last
The corresponding variable is called a leading variable.
Then
Any variable which is not a leading variable is a free variable:
|Leading Variables| + |Free Variables| = |Number of Columns of A|
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138. More Definitions
Definition
Suppose the augmented matrix for the linear system Ax = y has been
brought to echelon form.
If there is a leading 1 in any column except the last
The corresponding variable is called a leading variable.
Then
Any variable which is not a leading variable is a free variable:
|Leading Variables| + |Free Variables| = |Number of Columns of A|
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139. More Definitions
Definition
Suppose the augmented matrix for the linear system Ax = y has been
brought to echelon form.
If there is a leading 1 in any column except the last
The corresponding variable is called a leading variable.
Then
Any variable which is not a leading variable is a free variable:
|Leading Variables| + |Free Variables| = |Number of Columns of A|
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140. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
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141. First
We have
If the system is consistent and there are no free variables, then the solution
is unique.
Example
1 ∗ ∗ ∗
0 1 ∗ ∗
0 0 1 ∗
0 0 0 1
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142. First
We have
If the system is consistent and there are no free variables, then the solution
is unique.
Example
1 ∗ ∗ ∗
0 1 ∗ ∗
0 0 1 ∗
0 0 0 1
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143. Second
If the system is consistent and there are one or more free variables
There are infinitely many solutions.
Example
1 ∗ ∗ ∗
0 1 ∗ ∗
0 0 1 ∗
0 0 0 0
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144. Second
If the system is consistent and there are one or more free variables
There are infinitely many solutions.
Example
1 ∗ ∗ ∗
0 1 ∗ ∗
0 0 1 ∗
0 0 0 0
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145. Finally
The Last Case, we have a free variable, but a 1 in the last column
1 ∗ ∗
0 0 1
0 0 0
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146. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
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147. We have
Definition
A homogeneous system of linear algebraic equations is one in which all the
numbers on the right hand side are equal to 0:
a11x1 + a12x2 + ... + a1nxn =0,
a21x1 + a22x2 + ... + a2nxn =0,
· · · · · · · · · · · · · · · · · · · · · · · ·
am1x1 + am2x2 + ... + amnxn =0.
Remark
The homogeneous system Ax = 0 always has the solution x = 0.
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148. We have
Definition
A homogeneous system of linear algebraic equations is one in which all the
numbers on the right hand side are equal to 0:
a11x1 + a12x2 + ... + a1nxn =0,
a21x1 + a22x2 + ... + a2nxn =0,
· · · · · · · · · · · · · · · · · · · · · · · ·
am1x1 + am2x2 + ... + amnxn =0.
Remark
The homogeneous system Ax = 0 always has the solution x = 0.
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149. Thus
Non-Trivial Solutions
Any non-zero solutions to Ax = 0, if they exist, are called non-trivial
solutions.
We can use the Gauss-Jordan Algorithm
Reducing (A|0)
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150. Thus
Non-Trivial Solutions
Any non-zero solutions to Ax = 0, if they exist, are called non-trivial
solutions.
We can use the Gauss-Jordan Algorithm
Reducing (A|0)
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151. Example
You have the following homogeneous system
(A|0) =
1 2 0 −1 0
−2 −2 4 5 0
2 4 0 −2 0
Therefore, we have after reduction
1 2 0 −1 0
0 1 4 3 0
0 0 0 0 0
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152. Example
You have the following homogeneous system
(A|0) =
1 2 0 −1 0
−2 −2 4 5 0
2 4 0 −2 0
Therefore, we have after reduction
1 2 0 −1 0
0 1 4 3 0
0 0 0 0 0
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153. Given that the column of zeros does not change
After row operations, we will use
1 2 0 −1
0 1 4 3
0 0 0 0
Therefore
We have that
1 Leading variables x1, x2
2 Free variables x3, x4
Since there are no leading entries in the third or fourth columns.
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154. Given that the column of zeros does not change
After row operations, we will use
1 2 0 −1
0 1 4 3
0 0 0 0
Therefore
We have that
1 Leading variables x1, x2
2 Free variables x3, x4
Since there are no leading entries in the third or fourth columns.
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155. We can re-write
As echelon reduced form
1 0 −8 −7
0 1 4 3
0 0 0 0
Therefore, we have, using the free variable ideas
x1 = 8s + 7t
x2 = −4s − 3t
x3 = s
x4 = t
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156. We can re-write
As echelon reduced form
1 0 −8 −7
0 1 4 3
0 0 0 0
Therefore, we have, using the free variable ideas
x1 = 8s + 7t
x2 = −4s − 3t
x3 = s
x4 = t
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157. Therefore
We have in vector format
xH =
x1
x2
x3
x4
= s
8
−4
1
0
+ t
7
−3
0
1
|∀s, t ∈
80 / 96
158. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
81 / 96
159. We have
Basic Properties for a Am×n
1 The number of leading variables is ≤ min (m, n)
2 The number of non-zero equations in the echelon form of the system
is equal to the number of leading entries.
3 The number of free variables plus the number of leading variables
= n, the number of columns of A.
4 The homogeneous system Ax = 0 has non-trivial solutions if and only
if there are free variables.
5 A homogeneous system of equations is always consistent (At least a
solution)
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160. We have
Basic Properties for a Am×n
1 The number of leading variables is ≤ min (m, n)
2 The number of non-zero equations in the echelon form of the system
is equal to the number of leading entries.
3 The number of free variables plus the number of leading variables
= n, the number of columns of A.
4 The homogeneous system Ax = 0 has non-trivial solutions if and only
if there are free variables.
5 A homogeneous system of equations is always consistent (At least a
solution)
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161. We have
Basic Properties for a Am×n
1 The number of leading variables is ≤ min (m, n)
2 The number of non-zero equations in the echelon form of the system
is equal to the number of leading entries.
3 The number of free variables plus the number of leading variables
= n, the number of columns of A.
4 The homogeneous system Ax = 0 has non-trivial solutions if and only
if there are free variables.
5 A homogeneous system of equations is always consistent (At least a
solution)
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162. We have
Basic Properties for a Am×n
1 The number of leading variables is ≤ min (m, n)
2 The number of non-zero equations in the echelon form of the system
is equal to the number of leading entries.
3 The number of free variables plus the number of leading variables
= n, the number of columns of A.
4 The homogeneous system Ax = 0 has non-trivial solutions if and only
if there are free variables.
5 A homogeneous system of equations is always consistent (At least a
solution)
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163. We have
Basic Properties for a Am×n
1 The number of leading variables is ≤ min (m, n)
2 The number of non-zero equations in the echelon form of the system
is equal to the number of leading entries.
3 The number of free variables plus the number of leading variables
= n, the number of columns of A.
4 The homogeneous system Ax = 0 has non-trivial solutions if and only
if there are free variables.
5 A homogeneous system of equations is always consistent (At least a
solution)
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164. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
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165. We have
Theorem
If x is a solution to Ax = 0, then so is cx for any real number c.
Proof: Quite simple
Theorem
If x and y are two solutions to the homogeneous equation, then so is
x + y.
Proof
It is also simple!!!
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166. We have
Theorem
If x is a solution to Ax = 0, then so is cx for any real number c.
Proof: Quite simple
Theorem
If x and y are two solutions to the homogeneous equation, then so is
x + y.
Proof
It is also simple!!!
84 / 96
167. We have
Theorem
If x is a solution to Ax = 0, then so is cx for any real number c.
Proof: Quite simple
Theorem
If x and y are two solutions to the homogeneous equation, then so is
x + y.
Proof
It is also simple!!!
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168. Superposition Principle
We have
These two properties constitute the famous principle of superposition
which holds for homogeneous systems.
Restating
If x and y are two solutions to the homogeneous equation Ax = 0, then
any linear combination of x and y is also a solution.
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169. Superposition Principle
We have
These two properties constitute the famous principle of superposition
which holds for homogeneous systems.
Restating
If x and y are two solutions to the homogeneous equation Ax = 0, then
any linear combination of x and y is also a solution.
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170. Outline
1 System of Linear Equations
Introduction
System of Linear Equations
Matrices and Their Operations
Using the Matrix Operations
Example
Going back to the problem
2 Elementary row operations
Introduction
Elementary Matrices
Properties of the Elementary Matrices
The Theorem for the Gauss-Jordan Algorithm
The Gauss-Jordan Algorithm
Application to the solutions of Ax = y
Consistency and Inconsistency
3 Homogeneous and In-Homogeneous Systems
Homogeneous systems
Basic Properties
Linear combinations and the superposition principle
Inhomogeneous Systems
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171. We have
Definition
The system Ax = y is inhomogeneous if it is not homogeneous.
Example
x1 + 2x2 − x4 = 1
−2x1 − 3x2 + 4x3 + 5x4 = 2
2x1 + 4x2 − 2x4 = 3
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172. We have
Definition
The system Ax = y is inhomogeneous if it is not homogeneous.
Example
x1 + 2x2 − x4 = 1
−2x1 − 3x2 + 4x3 + 5x4 = 2
2x1 + 4x2 − 2x4 = 3
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173. Example
Augmented Matrix
(A|y) =
1 2 0 −1 1
−2 −3 4 5 2
2 4 0 −2 3
The row echelon form of the augmented matrix is
1 2 0 −1 1
0 1 4 3 4
0 0 0 0 1
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174. Example
Augmented Matrix
(A|y) =
1 2 0 −1 1
−2 −3 4 5 2
2 4 0 −2 3
The row echelon form of the augmented matrix is
1 2 0 −1 1
0 1 4 3 4
0 0 0 0 1
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175. Example
The reduced echelon form is
1 0 −8 −7 0
0 1 4 3 0
0 0 0 0 1
Problem, the third equation now reads
0x1 + 0x2 + 0x3 + 0x4 = 1
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176. Example
The reduced echelon form is
1 0 −8 −7 0
0 1 4 3 0
0 0 0 0 1
Problem, the third equation now reads
0x1 + 0x2 + 0x3 + 0x4 = 1
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177. Reasoning
If the original system has a solution
Then performing elementary row operations will give us an equivalent
system with the same solution.
But this equivalent system of equations is inconsistent
So the original system is also inconsistent.
In General
If the echelon form of (A|y) has a leading 1 in any position of the last
column, the system of equations is inconsistent.
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178. Reasoning
If the original system has a solution
Then performing elementary row operations will give us an equivalent
system with the same solution.
But this equivalent system of equations is inconsistent
So the original system is also inconsistent.
In General
If the echelon form of (A|y) has a leading 1 in any position of the last
column, the system of equations is inconsistent.
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179. Reasoning
If the original system has a solution
Then performing elementary row operations will give us an equivalent
system with the same solution.
But this equivalent system of equations is inconsistent
So the original system is also inconsistent.
In General
If the echelon form of (A|y) has a leading 1 in any position of the last
column, the system of equations is inconsistent.
90 / 96
180. Then
It is not true that any inhomogeneous system with the same matrix A
is inconsistent
It depends completely on the particular y which sits on the right hand side
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184. Therefore
We have
x1 = 8s + 7t − 7
x2 = −4s − 3t + 4
x3 = s
x4 = t
It is similar to the homogeneous system
x1 = 8s + 7t
x2 = −4s − 3t
x3 = s
x4 = t
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185. Therefore
We have
x1 = 8s + 7t − 7
x2 = −4s − 3t + 4
x3 = s
x4 = t
It is similar to the homogeneous system
x1 = 8s + 7t
x2 = −4s − 3t
x3 = s
x4 = t
93 / 96
187. Thus, by fixing
For example, s = t = 0
xp =
−7
4
0
0
The general solution to the inhomogeneous system
xI = xH + xp
95 / 96
188. Thus, by fixing
For example, s = t = 0
xp =
−7
4
0
0
The general solution to the inhomogeneous system
xI = xH + xp
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189. Finally
Theorem
Let x1
p and x2
p be two solutions to Ax = y. Then their difference
x1
p − x2
p is a solution to the homogeneous equation Ax = 0.
The general solution to Ax = y can be written as xI = xH + xp
where xH denotes the general solution to the homogeneous system.
96 / 96