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 rece
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 rec
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 map
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 rece
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
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 b
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 =
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
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
M346 Second Midterm Exam Solutions, November 9, 2004
22
.
1. Let A =
13
a) Find the eigenvalues and eigenvectors of A.
Since the sum of each row is 4, 1 = 4. Since the trace is 5, the other
eigenvalue
M346 First Midterm Exam Solutions, September 21, 2004
1. Let V be the subspace of R4 dened by the equation x1 + x2 + x3 + x4 = 0.
a) Find the dimension of V .
V is the null space of the rank-1 matrix
M346 First Midterm Exam, September 18, 2003
The exam is closed book, but you may have a single hand-written 8.5 11
crib sheet. There are 5 problems, each worth 20 points. The rst four are
calculationa
M346 Second Midterm Exam Solutions, October 23, 2003
1. Find all the eigenvalues of the following matrices. You do NOT need to
nd the corresponding eigenvectors. [Note: the answers are fairly simple,
M346 Second Midterm Exam Solutions, October 22, 2013
1) Find the eigenvalues of the matrix
5
3 00
3 1 0 3
.
A=
0
0 3 1
0
0 43
For each eigenvalue, indicate what the geometric and algebraic multiplicit
M346 First Midterm Exam Solutions, September 26, 2013
1) Consider the matrix
1212
A = 2 4 3 5 .
3 6 7 10
Find the reduced row-echelon form Arref , nd a basis for the null space of
A, and nd a basis fo
M346 Final Exam, December 11, 2013
1234
1. Basic row operations. Consider the matrix A = 1 3 5 8 .
2 5 8 12
a) Find a basis for the null space of A.
1 0 1 4
4 , which gives the equations
A row-reduces
M346 Final Exam, May 19, 2009
1. For each of these collections B of vectors in a vector space V , indicate (with
explanation) whether B is linearly independent, spans V , both, or neither.
1
3
2
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 =
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
M346 First Midterm Exam, July 25, 2011
1113
1) (25 pts) Consider the matrix A = 1 2 0 5 and the vector
2318
0
b = 1 (in R3 ). a) Find all solutions to Ax = b.
1
1113|0
1 0 2 1 | 1
Row-reducing 1 2 0 5
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 b
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 =
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 d
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
M346 (56615), Homework #13
Due: 03:30pm, Tuesday, May 07
Innite-dimensional inner product spaces (6.8)
A)
) with an = (1)n/ n . Is v in l2(R)?
i. Let v = (a1, a2, a3,
ii. Consider the function f (x) =
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. N
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
.
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 t
M346 FALL 2017 UNIQUE NUMBER 54355 HOMEWORK 10
This homework is due on Thursday, November 9, 2017, before lecture begins.
Late homework is not accepted. In order to receive credit for this assignment
M346 FALL 2017 UNIQUE NUMBER 54355 HOMEWORK 8
This homework is due on Thursday, October 26, 2017, before lecture begins.
Late homework is not accepted. In order to receive credit for this assignment y