Exercises on projections
Instructor: MasoodulAlam
1. (8. p214) Project the vector b = (1, 1) onto the lines a1 =
(1, 0) and a2 = (1, 2). Draw the projections p1 and p2 and add
p1 + p2 . Why is b , p1 + p2 ?
Answer.
p1 is obviously a1 .
a2
p2
"
1
p2 =
aT
Linear Algebra Assignment VII
Instructor: MasoodulAlam
Answer the questions showing the necessary steps.
2 1 0
stating clearly the singular
1. Find the svd of A =
0 1 2
values of A, and U, V, . Find the best rank one approximation
to A.
x
y
2.
Linear Algebra Assignment V
Instructor: MasoodulAlam
Answer the questions showing the necessary steps.
1 1 1
0 0 1
. Find the QR decomposition of A with R
1. Let A =
0 1 1
0 0 1
as a square matrix.
2. Rotate the line 4x 3y = 3 about the origin through
Linear Algebra Assignment IV
Instructor: MasoodulAlam
Answer the questions showing the necessary steps.
1. A system of linear equations cannot have exactly two solutions. Why?
(a). If (x,y,z) and (X,Y,Z) are two solutions, what is another
solution? (b).
Linear Algebra Assignment VI
Instructor: MasoodulAlam
Answer the questions showing the necessary steps.
1 2 2
1. Diagonalize the symmetric matrix A = 2 1 2 .
2 2 1
1 2 2
2. Diagonalize A = 2 1 2 using congruence.
2 2 1
Hints:
can
use elementary colu
Linear Algebra Assignment III
Instructor: MasoodulAlam
Answer the questions showing the necessary steps.
3 0
1. Describe geometrically all linear combinations of 2 , 4
5 0
3
and 0 . If possible find a vector not a linear combination of
5
th
Linear Algebra Assignment IX
Instructor: MasoodulAlam
Answer the questions showing the necessary steps.
1. Suppose
V is the vector space of all 2 2 real matrices M. Let
1 2
and T be the linear transformation T M = AM.
A =
3 4
Show that the kernel of
Linear Algebra Assignment VIII
Instructor: MasoodulAlam
Answer the questions showing the necessary steps.
1. Find the range and the kernel of the linear transformation T :
R3 R3 taking (x1 , x2 , x3 ) 7 (x1 , x1 , x1 ).
2. Is T : R3 R2 taking (x1 , x2 ,
Linear Algebra Assignment I
Instructor: MasoodulAlam
Question nos. 3, 6 and 8 will not be graded. No need to submit the solutions of these three questions. These are only for practice. Answer the remaining
questions showing the necessary steps.
1. Compu
Linear Algebra Assignment X
Instructor: MasoodulAlam
Answer the questions showing the necessary steps.
00
0
1. Solve the
ode y 3y 4y = 0 with the initial condition
y(0)
u(0) = 0 =
y (0)
Linear Algebra
15.1
Active and Passive rotation
Change of the coordinate system is called a passive transformation. A passive transformation is simply recoordination. In physical problems we may need to consider transformations in which the
vector itself
Linear Algebra
22
22.1
More on vectors and geometry in 3space
Direction cosines
The direction cosines of a vector v in 3space are the numbers cos , cos , cos ,
where , and are the angles between v and the positive x, y and
z axes.
[Recall the angle betw
Linear Algebra
The Method of Least Squares
Suppose we are given m points (ti , bi ), i = 1, . . . , m in the tbplane
and are asked to fit a line b = C + Dt to these points.
[Why are we using variables t, b in stead of familiar x, y will be clear
soon.]
I
Linear Algebra
6
Matrix Algebra
Two matrices are equal if all their respective entries are equal. In
particular they have the same dimension. Thus they have the same
number of rows and the same number of columns.
Ex.
a b e f
=
a = e, b = f, c = g an
Linear Algebra
12.8
Row space and the left nullspace
The row space of an mn matrix A is the column space of AT . Since
AT is n m, the row space of A is a subspace of Rn .
The row space consists of all linear combinations of the rows.
The left nullspace of
Linear Algebra
10.8.1
Homogeneous linear systems
A system of linear equations for which the constant terms are zero is
call a homogeneous system. A homogeneous system is of the form
AX = 0.
An m n linear system AX = b of equations, homogeneous or not,
can
Linear Algebra
1
Introduction
To understand the importance of Linear Algebra we begin with the
language of system. We give an input to the system and get an
output from it. We add two inputs and put their sum in the system as
input. If the resulting outpu
Linear Algebra
25.2
Factorization and Change of basis
We can achieve following factorizations of matrices by change of
basis. We suppose the matrices are real.
1. SVD. A = UV T . We consider Amn and to represent the same
linear transformation T : Rn Rm re
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