MATH 2270 Summer 2016
August 4, 2016
The following are problems from the textbook Linear Algebra and its Applications, Fifth Edition ; by David Lay, Steven Lay, and Judi McDonald; ISBN: 978-0321982384. Suggested problems from each section will be added as we make progress in class. Sometimes I might formulate problems on my own or use a different source. Feel free to come to office hours if you are stuck at any of the problems. If you want to practice more problems, try the other problems in the textbook.
HW is not to be turned in.
1 Recommended problems for quiz #1
• Section 1.1: 11, 7, 15, 18, 19, 23, 24
• Section 1.2: 1, 3, 5, 13, 15, 21, 22, 23-32 (Most of these are conceptual.)
• Section 1.3: 5, 9, 11, 13, 15, 18, 21, 22, 23, 24, 25
• Section 1.4: 1, 5, 9, 13, 15, 23, 24, 29, 32
2 Recommended problems for test #1
• Same as for quiz #1
• Section 1.5: 23, 24, 27, 28, 31, 33, 35, 37
• Section 1.7: 21, 22, 26, 27, 28, 31, 37, 38
3 Recommended problems for quiz # 2
• Section 1.8: 1, 3, 7, 11, 13, 17, 19, 21, 22, 29, 30
• Section 1.9: 5, 7, 11, 15, 17, 21, 23, 24, 25, 29, 30, 31, 32, 35
4 Recommended problems for quiz # 3
• Section 2.1: 5, 10, 11, 12, 15, 16, 19, 21, 23, 24, 25, 27, 31
• Section 2.2: 1, 5, 9, 10, 11, 13, 15, 21, 22, 24, 31, 37
• Section 2.3: 3, 11, 12, 13, 17, 19, 21, 27, 31, 33, Problems 15-24 are based on Invertible Matrix
Theorem
• Extra problem: Prove the following statement.
A~ =
~b
= 0 has atmost one solution for every
~b if and only if A~ =
• Extra problem: Let
Similarly T
β
T
α
:
R
2 →
R be rotation by angle β .
2 be rotation by angle α counterclockwise about the origin.
– Write the standard matrix for T
α and T
β
.
1

Instructor : Radhika Gupta Math 2270 Summer 2016
– Compute the standard matrix of the transformation T
β rotations.
◦ T
α
, which is a composition of the two
– Compute the standard matrix of T
α
◦ T
β
.
– Describe geometrically the compositions T
β
◦ T
α and T
α
◦ T
β
.
– Explain geometrically why T
α
◦ T
β
= T
α
◦ T
β
.
– Show that the standard matrix of T
α
◦ T
β is the same as that of T
α + β
, rotation by α + β .
5 Recommended problems for test #2
• Same as for quiz #2, quiz #3
• Section 3.1: 1, 3, 9, 22, 37, 41
• Section 3.2: 1, 3, 9, 13, 15, 19, 21, 25, 29, 31-36, 39(b, d, e)
• Section 3.3: 23, 24, 25, 27
• Section 4.1: 1, 2, 7, 9, 11, 13, 15, 23, 24, 21 (this problem has another example of a vector space, but we didn’t talk about it in class)
• Section 4.2: 1, 5, 17, 21, 23, 25, 26, 31, 33 (extra)
6 Recommended problems for quiz # 4
• Section 4.3: 1, 5, 7, 9, 13, 15, 19, 25 and Practice Problem 3
• Section 4.4: 1, 3, 5, 7, 9, 11, 13, 27, 31, 32, 21 (optional - it is a preview for Section 4.7)
7 Recommended problems for quiz #5
• Section 4.5: 1, 5, 9, 11, 13, 21, 23
• Section 4.6: 1, 3, 5, 9, 13, 15, 25, 27, 17, 18 (T/F)
• Section 4.7: 1, 5, 7, 9, 13, 14
8 Recommended problems for test #3
• Same as for quiz #4, quiz #5
• Section 4.9: 1, 3, 11, 13
• Section 5.1: 3, 5, 9, 11, 13, 24, 25, 27, 29, 31, 32, 21, 22 (T/F), Practice Problem #2, 4
• Section 5.2: 1, 3, 5, 9, 11, 15, 17, 19, (21,22 - T/F)
9 Recommended problems for quiz #6
• Section 5.3: 1, 5, 7, 9, 11, 13, 15, 23, 31, 32
• Section 5.4: 5, 9, 11, 15, 17
• Section 5.6: 1, 3, 7, 9, 11, 13, 17(a,b)
10 Recommended problems for quiz #7
• Section 6.1: 1, 3, 5, 7, 11, 13, 17
• Section 6.2: 1, 5, 7, 9, 13, 17
• Section 6.3: 1, 3, 5, 7, 11, 13, 15, 17
2

Instructor : Radhika Gupta Math 2270 Summer 2016
11 Recommended problems for test #4
• Same as for quiz #6, quiz #7
• Section 6.4: 1, 3, 5, 7, 9, 11
• Section 6.5: 1, 3, 5, 7, 9, 11
• Section 6.6: 1, 2, 3, 4, Some extra problems which will not be tested - 7, 9
• Section 7.1: 1, 5, 6, 9, 11, 13, 17, 21, 23
12 Recommended problems
• Section 7.4: 1, 15, 16
13 Problems on Spectral Decomposition and SVD
(a) Let A =

3 − 2 4


− 2 6 2

4 2 3 with an orthogonal diagonalization given by


√
2
0

√
2
− 1
√
18
4
18
√
1
18
− 2
3
− 1
3
2
3





 7 0 0

0 7 0
0 0 − 2



√
2
0
√
2
− 1
√
18
4
18
√
1
18
− 2 
T
3
− 1

3
2

3
Construct a spectral decomposition of A . (Look at Example 4 in Section 7.1)
 1 − 1 
(b) Let the singular value decomposition of a matrix A =

− 2 2
3 3
 be given as follows:


0
0
− 1
2
5
√
5
1 0
√
5

√
5
0

 3

√
2
√
0
0

10

" 1
2
0 0
√
2
− 1
1
2
√
2
#
T
• What is the rank of A ?
• Write an orthonormal basis for Row( A ) and Col( A ).
• Find Nul( A ).
• Write the best approximation for A by one rank 1 matrix. (given by σ
1 u
1
• Write the best approximation for A by two rank 1 matrix. (given by σ
1 u
1
T
1
)
T
1
+ σ
2 u
2
T
2
)
3