M346 (56615), Homework #3
Due: 3:30pm, Tuesday, Feb. 05
Instructions: Questions are from the book Applied Linear Algebra, 2nd ed. by Sadun. Please
show all your work, not only your nal answer, to receive credit. Keep answers organized in the
same order th
M346 (56615), Homework #3 Solutions
* = graded
Linear transformations and operators (3.1)
p. 41-42:
3. The integral
tion is linear, I
11
6. L = 2 0
1 4
of a continuous function is continuous, so I maps C [0, 1] to itself. Since integrais a linear operator
M346 (56615), Homework #6
Due: 3:30pm, Thursday, Mar. 07
Instructions: Questions are from the book Applied Linear Algebra, 2nd ed. by Sadun. Please
show all your work, not only your nal answer, to receive credit. Keep answers organized in the
same order t
M346 (56615), Midterm #2 Solutions
Question #1 (25 points)
52 4
Let A = 0 3 1 .
00 3
a) Find the eigenvalues of A and nd a basis for their corresponding eigenspaces. [Hint:
There is a very quick way to determine the eigenvalues of this matrix.]
Solution:
M346 (56615)
Applied Linear Algebra, Spring 2013
Course syllabus (last revised: 03/29/2013)
Instructor: Ravi Srinivasan
Email: rav@math.utexas.edu
Office: RLM 11.164
Office hours: M 3:30-5:30 (tentative, will change depending on HW due dates)
Lecture: TTh
M346 (56615), Final Exam Solutions
Question #1 (25 points)
Matrix representation/change of basis/kernel and range.
x1 x2
: xi R be the space of 2 2 real symmetric matrices with standard basis
x2 x3
00
01
10
. The linear operator L: V V is dened by
,
,
01
M346 Final Exam Solutions, December 11, 2004
1. On R3 [t], let L be the linear operator that shifts a function to the left by
one. That is (Lp)(t) = p(t + 1). Find the matrix of L relative to the standard
basis cfw_1, t, t2 , t3
Solution: Note that L(b1
M346 Second Midterm Exam Solutions, August 5, 2011
42
1) (25 pts) Consider the matrix A =
.
8 2
a) Find the eigenvalues of A. For each eigenvalue, nd a corresponding
eigenvector.
The sum of each row is 6, the trace is 2, and the determinant is -24. From
a
M346 Final Exam Solutions, August 15, 2011
123
1
100 2
4 7 1 24
row-reduces to B = 0 1 0 5 .
1) The 44 matrix A =
1 1 4
0 0 1 3
9
4 8 1 29
000 0
Let ai denote the ith column of A.
a) Write one of the columns of A explicitly as a linear combination of
M346 Second Midterm Exam Solutions, April 7, 2011
01
1) The matrix A =
has eigenvalues 1 = 1 and 2 = 4, with
4 3
1
1
. Suppose that x(n) satises the
and b2 =
eigenvectors b1 =
4
1
system of equations x(n + 1) = Ax(n) for all n 0.
a) If x(0) is random (mea
M346 First Midterm Exam Solutions, February 17, 2011
3
1
1
1) (15 points) Consider the vectors 1 , 2 and 1 in R3 . Are these
5
5
3
3
vectors linearly independent? Do they span R ? Do they form a basis for
R3 ?
113
Answer: Row-reducing the matrix A = 1
M346 Final Exam Solutions, May 14, 2011
5
10
. Con15 20
120
1
2
.
and the vector x =
,
sider the basis B =
70
3
1
a) Find the coordinates of x in the B basis. (That is, nd [x]B .)
3 1
21
1
, so [x]B = PBE x =
, so PBE = PEB = 1
PEB =
5
1 2
13
1
2
120
58
.
M346 Final Exam, December 15, 2009
1325
1 0 0 4/11
2 1 1 1
0 1 0 13/11
1) The matrix A =
row-reduces to B =
.
0
0 0 1 10/11
1 2 3
3357
000
0
a)Find all solutions to Ax = 0.
These are the same as the solutions to B x = 0, namely all multiples of
(
M346 First Midterm Exam Solutions, February 11, 2009
1
0
0
1a) In R3 , let E be the standard basis and let B = 2 , 1 , 0 be
1
4
3
3
an alternate basis. Let v = 2 . Find PEB , PBE and [v]B .
14
100
1
00
1
PEB = (b1 b2 b3 ) = 2 1 0 , PBE = PEB = 2 1 0 ,
M346 (56615), Midterm #1 Solutions
Question #1 (25 points)
Let V be the space of real 2 2 skew-symmetric matrices (i.e., those of the form
10
00
a, b, d R). Equip V with standard basis E =
00
01
12
.
,
,
01
1 4
2 3
,
01
1 0
,
00
01
ab
b d
with
, and dene
M346 (56615), Homework #13 Solutions
* = graded
Innite-dimensional inner product spaces (6.8)
A)
2
i. No, v is not in l2(R) since v
x1 2 p
1 2p
| f (x)|2dx =
x 2 pdx =
1
1
n =1 n
|an |2 =
= + .
1/2. Note that
ii. First, we consider the case p
n =1
=
1
x=
M346 (56615), Homework #7
Due: 3:30pm, Thursday, Mar. 21
Instructions: Questions are from the book Applied Linear Algebra, 2nd ed. by Sadun. Please
show all your work, not only your nal answer, to receive credit. Keep answers organized in the
same order t
M346 (56615), Homework #7 Solutions
* = graded
Long-time behavior and stability (5.5)
p. 125:
*2. The eigenvalues are 1 = 1 and 2 = 2 with corresponding eigenvectors b1 = (2, 1)T and
b2 = (1, 1)T . Therefore, b1 is a neutrally stable mode while b2 is unst
M346 (56615), Homework #8
Due: 3:30pm, Thursday, Mar. 28
Applications to network science
Consider the following directed network N :
i. Determine the adjacency matrix A of the network N .
ii. What are the in-degrees din and out-degrees dout of each node i
M346 (56615), Homework #9
Due: 3:30pm, Tuesday, Apr. 09
Instructions: Questions are from the book Applied Linear Algebra, 2nd ed. by Sadun. Please
show all your work, not only your nal answer, to receive credit. Keep answers organized in the
same order th
M346 (56615), Homework #10
Due: 3:30pm, Tuesday, Apr. 16
Instructions: Questions are from the book Applied Linear Algebra, 2nd ed. by Sadun. Please
show all your work, not only your nal answer, to receive credit. Keep answers organized in the
same order t
M346 (56615), Homework #11
Due: 03:30pm, Tuesday, Apr. 23
Self-adjoint and normal operators
030
A) Find an orthonormal basis consisting of eigenvectors of A = 3 0 4 .
040
2 0i
B) Find an orthonormal basis consisting of eigenvectors of A = 0 1 0 .
i 0 2
C)
M346 (56615), Homework #11 Solutions
* = graded
Self-adjoint and normal operators
A) The eigenvalues of A are 1 = 5, 2 = 0, and 3 = 5 with corresponding eigenvectors e1 =
1
1
1
T
T
T
(3, 5, 4) , e2 = (4, 0, 3) , and e3 = (3, 5, 4) . Since A is self-adjoi
M346 (56615), Homework #12
Due: 03:30pm, Tuesday, Apr. 30
Singular value decomposition (SVD)
A) Show that if A is positive, its spectral decomposition A = U DU agrees exactly with its
singular value decomposition A = U V (i.e., show that = D and V = U ).
M346 (56615), Homework #12 Solutions
* = graded
Singular value decomposition (SVD)
A) If A is positive, then AA = A2 = A and AA = A2 = A. Therefore, the singular
values of A exactly the eigenvalues of A and the eigenvectors of AA are the same
are
as those