This document defines vectors and vector spaces. It begins by defining vectors in 2D and 3D space as matrices and describes operations like addition, scalar multiplication, and subtraction. It then defines a vector space as a set of vectors that satisfies 10 axioms related to these operations. Examples of vector spaces include the set of 2D and 3D vectors, sets of matrices, and sets of polynomials. The document also defines subspaces and proves that the span of a set of vectors in a vector space forms a subspace.
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
Vector Space & Sub Space Presentation
Presented By: Sufian Mehmood Soomro
Department: (BS) Computer Science
Course Title: Linear Algebra
Shah Abdul Latif University Ghotki Campus
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.
2. Linear Algebra for Machine Learning: Basis and DimensionCeni 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 second part which is discussing basis and dimension.
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
Chapter 12
Section 12.1: Three-Dimensional Coordinate Systems
We locate a point on a number line as one coordinate, in the plane as an ordered pair, and in
space as an ordered triple. So we call number line as one dimensional, plane as two
dimensional, and space as three dimensional co – ordinate system.
In three dimensional, there is origin (0, 0, 0) and there are three axes – x -, y - , and z – axis. X –
and y – axes are horizontal and z – axis is vertical. These three axes divide the space into eight
equal parts, called the octants. In addition, these three axes divide the space into three
coordinate planes.
– The xy-plane contains the x- and y-axes. The equation is z = 0.
– The yz-plane contains the y- and z-axes. The equation is x = 0.
– The xz-plane contains the x- and z-axes. The equation is y = 0.
If P is any point in space, let:
– a be the (directed) distance from the yz-plane to P.
– b be the distance from the xz-plane to P.
– c be the distance from the xy-plane to P.
Then the point P by the ordered triple of real numbers (a, b, c), where a, b, and c are the
coordinates of P.
– a is the x-coordinate.
– b is the y-coordinate.
– c is the z-coordinate.
– Thus, to locate a point (a, b, c) in space, start from the origin (0, 0, 0) and move a
units along the x-axis. Then, move b units parallel to the y-axis. Finally, move c
units parallel to the z-axis.
The three dimensional Cartesian co – ordinate system follows the right hand rule.
Examples:
Plot the points (2,3,4), (2, -3, 4), (-2, -3, 4), (2, -3, -4), and (-2, -3, -4).
The Cartesian product x x = {(x, y, z) | x, y, z in } is the set of all ordered triples of
real numbers and is denoted by 3 .
Note:
1. In 2 – dimension, an equation in x and y represents a curve in the plane 2 . In 3 –
dimension, an equation in x, y, and z represents a surface in space 3 .
2. When we see an equation, we must understand from the context that it is a curve in the
plane or a surface in space. For example, y = 5 is a line in 2 �but it is a plane in 3 �
������
3. in space, if k, l, & m are constants, then
– x = k represents a plane parallel to the yz-plane ( a vertical plane).
– y = k is a plane parallel to the xz-plane ( a vertical plane).
– z = k is a plane parallel to the xy-plane ( a horizontal plane).
– x = k & y = l is a line.
– x = k & z = m is a line.
– y = l & z = m is a line.
– x = k, y = l and z = m is a point.
Examples: Describe and sketch y = x in 3
Example:
Solve:
Which of the points P(6, 2, 3), Q(-5, -1, 4), and R(0, 3, 8) is closest to the xz – plane? Which point
lies in the yz – plane?
Distance between two points in space:
We simply extend the formula from 2 to . 3 . The distance |p1 p2 | between the points
P1(x1,y1, z1) and P2(x2, y2, z2) is: 2 2 21 2 2 1 ...
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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
Linear Algebra and Differential Equations by Pearson - Chapter 4Chaimae Baroudi
Chapter 4 Vector Spaces, define vectors and their properties, an introduction to vector spaces, subspaces and subsets, in addition to Linear Combinations, Bases and Dimensions for Vector Spaces. It also shows how to find Row, Column and Null Spaces for a matrix.
Linear Algebra and Differential Equations by Pearson - Chapter 3Chaimae Baroudi
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Linear Algebra and Differential Equations by Pearson - Chapter 1Chaimae Baroudi
Chapter 1 First Order Differential Equations, evolving General and Particular Solutions to ODE, Slope Fields and solving Linear, Separable, Homogeneous, Bernoulli and Exact Equations.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Biological screening of herbal drugs: Introduction and Need for
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3. Vectors in the Plane and in 3-Space
Defn - A vector in the plane is a 2 x 1 matrix
where x and y are real numbers
Notation - In print, use bold letters for vectors.
When writing by hand use
Defn - The numbers x and y in the definition of a
vector are called the components of the vector v.
Defn - Two vectors and
are equal iff x1 = x2 and y1 = y2
x
y
v
v
1
1
x
y
v 2
2
x
y
u
4. Vectors in the Plane and in 3-Space
A two-dimensional vector has several geometric
interpretations
1)A point ( x,y ) in the plane
2)A directed line segment from the origin to the point ( x,y )
3)A directed line segment from the point ( x1,y1 ) to the
point ( x2,y2 ). Then x = x2 x1, y = y2 y1
For applications to geometry and physics, some of the most
important operations are
1)Vector addition
2)Scalar multiplication
3)Vector subtraction
x
y
v
5. Vectors in the Plane and in 3-Space
Vector Addition
Defn - Let and . Define
Geometric interpretation
1
2
u
u
u 1
2
v
v
v 1 1
2 2
u v
u v
u v
6. Vectors in the Plane and in 3-Space
Vector Addition
u + v can be determined geometrically by adding
vectors head to tail
x
y
u
v
u v
v
If u and v have the same
length (to be defined later),
then u + v lies along the
bisector of the angle
between u and v
7. Vectors in the Plane and in 3-Space
Scalar Multiplication
Defn - Let c be a scalar (e.g. a real number) and
be a vector. The scalar multiple c u of u by c is
defined as the vector
1
2
u
u
u
1
2
cu
cu
8. Vectors in the Plane and in 3-Space
Vector Subtraction
Define u v as u + ( 1) v
Geometric relationship among u,v, u + v and u v
9. Vectors in the Plane and in 3-Space
Parametric Description of a Line
Let P1 and P2 be points. The line passing through
them can be described as
1 2 1 1 21t t t tP P P P P P
P1
P2
P2 P1
t 0
t 1
t 1
t 0
0 t 1
10. Vectors in the Plane and in 3-Space
Three-Dimensional Vectors
A three-dimensional vector is a 3 x 1 matrix
where x, y and z are real numbers
Vector addition, scalar multiplication and vector
subtraction are defined analogously to two
dimensions
x
y
z
11. Vectors in the Plane and in 3-Space
Basic Properties of Vectors in R2 or R3
Theorem - If u, v and w are vectors in R2 or R3
and c and d are real scalars, then
a)u + v = v + u
b) u + (v + w) = (u + v) + w
c)u + 0 = 0 + u = u
d) u + ( u) = 0
e)c (u + v) = c u + c v
f)(c + d) u = c u + d u
g) c (d u) = (c d ) u
h) 1 u = u
12. Slide 4.1- 12
VECTOR SPACES AND SUBSPACES
Definition:
A vector space is a nonempty set V of objects,
called vectors,
on which are defined two operations,
called addition and multiplication by scalars
(real numbers),
subject to the ten axioms (or rules) listed below.
The axioms must hold for all vectors u, v, and w in V
and for all scalars c and d.
13. Vector Spaces
Axioms
1) If u, v Î∈ V, then
2) for all u, v, w Î∈ V
3) $∃ a unique 0 Î∈ V such that
4) "∀ u Î∈ V $∃ a unique v Î∈ V such that
5) "∀ u, v Î∈ V
6) "∀ u Î∈ V and "∀ real number c, Î∈ V
7) "∀ real numbers c and "∀ u, v Î∈ V,
8) "∀ real numbers c and d, "∀ u Î∈ V,
9) "∀ real numbers c and d, "∀ u Î∈ V,
10) "∀ u Î∈ V
u v V
u v w u v w
0 u u 0 u u V
u v v u 0
u v v u
c u
c u v c u c v
c d u c u d u
c d u cd u
1 u u
14. Vector Spaces
Comments
V is called a real vector space because scalars c
and d are real numbers, not because of any
characteristics of the elements of V, e.g. if the
elements of V involve complex numbers, but c and
d are real, then V is a real vector space
The identity element 0 for does not need to
have anything to do with the number zero
There is nothing in the definition that requires the
elements of V to be columns of numbers
15. Vector Spaces
Comments
To show that V is a vector space, we have to prove
properties 1 through 10. So, the verification
involves ten small proofs
To show that V is not a vector space, all that we
have to do is show that a single property does not
hold
19. Vector Spaces: Examples – 2D and 3D vectors
Example
The set of two-dimensional vectors, as defined
earlier, forms a vector space. The verification of
the properties follows from the properties of
matrices
Example
The set of three-dimensional vectors, as defined
earlier, forms a vector space. The verification of
the properties follows from the properties of
matrices
20. Vector Spaces: Examples – nD vectors & matrices
Example
The set Rn of all n x 1 matrices with real entries
forms a vector space with as matrix addition
and as scalar multiplication
Example
The set mRn of all real m x n matrices forms a
vector space with as matrix addition and as
scalar multiplication
1
2
n
x
x
x
21. Vector Spaces: Examples - Polynomials
Example
Consider polynomials in a variable t
where a0, a1, …, an are real numbers. If a0 ≠ 0, the
degree of p(t) is n. The zero polynomial
has no
degree. Polynomials of degree 0 are constants. Let
Pn be the set of all polynomials of degree ≤ n
together with the zero polynomial. Let p(t) be as
above and let
1
0 1 1
n n
nnp t a t a t a t a
1
0 0 0 0n n
t t t
1
0 1 1
n n
nnq t b t bt b t b
22. Vector Spaces : Examples - Polynomials
Example (continued)
Define as
For a real number c, define as
Pn forms a vector space
p t q t
1
0 0 1 1 1 1
n n
n nn np t q t a b t a b t a b t a b
c p t
1
0 1 1
n n
nnc p t ca t ca t ca t ca
23. Vector Spaces : Examples – Continuous fonctions
Example
Let V be the set of all real valued continuous
functions on the closed interval [ 0,1 ].
For f, g Î∈ V, define as
For f Î∈ V and a real number c, define as
V is a vector space.
It is commonly called C [ 0,1 ].
f g
0,1f g t f t g t t
c f
0,1c f t cf t t
24. Vector Spaces
Theorem - If V is a vector space, then
a) for every u in V
b) for every scalar c
c) If , then either c = 0 or u = 0
d) for every u in V
0 u 0
c 0 0
c u 0
1 u u