Vector Space & Sub Space Presentation
Presented By: Sufian Mehmood Soomro
Department: (BS) Computer Science
Course Title: Linear Algebra
Shah Abdul Latif University Ghotki Campus
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
In general, we can find the coordinates of a vector u with respect to a given basis B by solving ABuB = u, for uB, where ABis the matrix whose columns are the vectors in B. ABis called thechange of basis matrix for B.
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.
In general, we can find the coordinates of a vector u with respect to a given basis B by solving ABuB = u, for uB, where ABis the matrix whose columns are the vectors in B. ABis called thechange of basis matrix for B.
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.
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Linear algebra power of abstraction - LearnDay@Xoxzo #5Xoxzo Inc.
LearnDay@Xoxzo is a monthly online seminar initiated by the Xoxzo team. We will have speakers from the team or guest speakers which will talk for 20 minutes each, on a subject of their choosing.
Linear algebra power of abstraction by Akira.
XOXZO Learn day
2018/12/21
======================
We have recorded sessions of our previous LearnDay here: https://www.youtube.com/channel/UCiV-bQprArQxKBSzaKY1vQg
For updates and news on our future LearnDays, follow us on Twitter (https://twitter.com/xoxzocom/) or sign up for our Exchange Newsletter (https://info.xoxzo.com/en/exchange-mailing-list/)
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 �
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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 ...
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
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.
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.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
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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Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
2. 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.
3. VECTOR SPACES AND SUBSPACES
1. The sum of u and v, denoted by
u+v , is in V.
2. 2. u+v=v+u
3. 3.(u+v)+w=u+(v+w)
4. 4.There is zero vector 0 in v such
that u+0=u.
5.For each u in V, there is a vector -u in
V such that u+(-u)=0.
6.The scalar multiple of u by c,denoted
by cu is in V.
7.c(u+v)=cu+cv.
8.(c+d)u=cu+du.
9.c(du)=(cd)u.
10.lu=u.
Using these axioms, we can
show that the zero vector in
Axiom 4 is unique, and the
vector , called the negative
of u, in Axiom 5 is unique for
each u in V.
4. For each u in V
and scalar c,
0u=0
c0=0
-u=(-1)u
VECTOR SPACES AND SUBSPACES
5. VECTOR SPACES AND SUBSPACES
EXAMPLE OF VECTOR SPACE:
Determine whether the set of V of all
pairs of real numbers (x,y) with the
operations (x1,y1) + (x2,y2) = (x1+x2+1,
y1+y2+1) and k(x,y) = (kx,ky) is a vector
space.
6. Solution:
let u=(x1,y1), v=(x2,y2) and w=(x3,y3) are objects in V and k1,k2 are some scalars.
1 . u+v = (x1,y1) + (x2,y2) = (x1+x2+1, y1+y2+1) since x1+x2+1, y1+y2+1 are also
real numbers . Therefore, u+v is also an object in V.
2. u+v = (x1+x2+1, y1+y2+1) = (x2+x1+1, y2+y1+1) = v + u Therefore , vector
addition is commutative.
3. u+(v+w) = (x1,y1)+[(x2,y2) +(x3,y3)] = (x1,y1)+(x2+x3+1, y2+y3+1) = [x1+
(x2+x3+1)+1 , y1+(y2+y1+1)+1) = [(x1+ x2+1)+x3+1 , (y1+y2+1)+y3+1)] =
(x1+x2+1, y1+y2+1)+(x3+y3) = (u+v)+w Hence, vector addition is associative.
4. Let (a,b) be in object in V such that (a,b)+u=u (a,b) +(x1,y1)=(x1,y1)
(a+x1+1,b1+y1+1) = (x1,y1) a= -1 , b=-1 Hence, (-1,-1) is zero vector in V.
Let (a,b) be in object in V such that (a,b)+u=(-1,-1) (a,b)+(x1,y1)=(-1,-1)
(x1+a+1,y1+b+1)=(1,-1) a= -x1-2 , b = -y1-2 Hence, (-x1-2,y1-2) is the negative of
u in V.
VECTOR SPACES AND SUBSPACES
7. VECTOR SPACES AND SUBSPACES
Definition: A subspace of a vector space V
is a subset H of V that has three properties:
a.The zero vector of V is in H.
b.H is closed under vector addition.That is for each u & v in
H, the sum u+v is in H.
c.H is closed under multiplication by scalars.That is for
each u in H and each scalar c,the vector cu is in H.
8. VECTOR SPACES AND SUBSPACES
Properties (a), (b), and (c) guarantee that a subspace H of V is itself a vector
space, under the vector space operations already defined in V.
Every subspace is a vector.
Properties (a), (b), and (c) guarantee that a subspace H of V is itself a vector
space, under the vector space operations already defined in V.
Conversely, every vector space is a subspace (of itself and possibly of other
larger spaces).
9. VECTOR SPACES AND SUBSPACES
EXAMPLE OF SUBSPACE:
A SUBSPACE OF M 2x2
Let W be the set of all 2×2 symmetric
matrices. Show that W is a subspace of
the vector space M2×2, with the standard
operations of matrix addition and scalar
multiplication.
11. A SUBSPACE SPANNED BY A SET
The set consisting of only the zero vector in a
vector space V is a subspace of V, called the zero
subspace and written as {0}.
As the term linear combination refers to any sum
of scalar multiples of vectors, and Span {v1,…,vp}
denotes the set of all vectors that can be written as
linear combinations of v1,…,vp.
12. A SUBSPACE SPANNED BY A SET
Example 10: Given v1 and v2 in a vector space V,
let
. Show that H is a subspace of
V.
Solution: The zero vector is in H, since
.
To show that H is closed under vector addition, take
two arbitrary vectors in H, say,
and .
By Axioms 2, 3, and 8 for the vector space V,