Solutions to Exercises
44
Problem Set 3.1, page 131
Note An interesting max-plus vector space comes from the real numbers R combined
with . Change addition to give x + y = max(x, y) and change multipl
T
’ X I ‘ ‘ ’ “I
LIL' _ . ' "‘ ‘ . I. A . ' ' I. Q A‘ .i‘ A Al; x . A A L‘ I l' V‘ :
r I- I i ‘ t‘ J
JAl C vl \‘L “U I‘. m‘ ‘ A; . Ahuvthl} 1. £9; mtA' A I, .1
) V I l 1
. . In 4L v C "A a H o. ' -: a
:I‘U‘l , .‘J-m
V
I
.‘ ”v ‘>_ ‘ ; I r
'Q'CX‘U ‘Z—a ‘ A i LAL’L.‘ .' l.)- 'A."x !. tk' Al; \‘
A In
L“ ' ‘
a ,4.» I. . 1:. ‘ A! u, A #9 ~ '
{.A'
n_‘
I4
:‘L‘ WA
: I . 0 Wm
«Jaw n5. . A." W .s ,.
‘19. ’
hi
m
Du}
»
A
I,
h.
C
A.
n .w
a . -
\ ﬂ
A. .v A
A" 2 L.
L n m .
r a v w w _0
a n n W n
. M I i '
,. w ix L m
1 . ,m ‘ . A
("O
u
\
0A1, A A x. g,» ‘
t
it? I is in ,
I; | ~ ‘ , ‘ .
3 h l:- ’ an
Homework 6 - SOLUTION
Due: Tuesday, October 24, 2017, 10:00PM
(30 points total)
1.
(8 points total; 2 points for correct answer in each subpart). Are the following statements true or false?
Give reaso
Homework 7 - SOLUTION
Due: Tuesday, October 31, 2017, 10:00PM
(30 points)
1.
(3 points for completion) Find a formula for the eigenvalues of the 2 2 matrix
].
=[
What is the sum of the eigenvalues?
Homework 4 Solution
Due: Tuesday, October 10, 2017, 10:00PM
30 points total
1. (10 points total; 2 points each subpart, for correctness) Construct a matrix with the desired
property or say why that is
Homework 8 - SOLUTION
Due: Tuesday, November 14, 2017, 10:00PM
(30 points total)
1. ( 9 points total; 3 points for completion in each subpart) Compute the SVD decomposition =
of the following matrice
‘ t r ) '
.‘.I .' , ‘ LII. , ,l,_ an A'L a .-‘ x. .r Crn'ﬁ'u
A . A at! ._ .u ML
i
n
C
I ’ L I .‘ “b
\ r 3
K b , ‘ ' _ I“ m - _ > ’ Ii l ”‘l , In, ‘ ll. ' ‘ V Plants ‘ J _ . ‘ or“
, l.,"'
h .‘i. A,
218
Chapter 4. Orthogonality
4.3 Least Squares Approximations
It often happens that Ax D b has no solution. The usual reason is: too many equations.
The matrix has more rows than columns. There are mo
Solutions to Exercises
69
Problem Set 4.1, page 202
1 Both nullspace vectors will be orthogonal to the row space vector in R3 . The column
space of A and the nullspace of AT are perpendicular lines in
Solutions to Exercises
98
Problem Set 6.1, page 298
1 The eigenvalues are 1 and 0.5 for A, 1 and 0.25 for A2 , 1 and 0 for A . Exchanging
the rows of A changes the eigenvalues to 1 and 0.5 (the trace
Linear Algebra - Fall 2015
HW #8
Please give
Section
Section
Section
complete, well-written solutions to the following exercises from Strang.
4.2: 1, 3, 13, 17, 30
4.3: 10, 12, 21
4.4: 5, 10
1
Linear Algebra - Fall 2015
HW #10
Please give
Section
Section
Section
Section
Section
Section
Section
Section
complete, well-written solutions to the following exercises from Strang.
6.1: 19, 34
6.2:
Linear Algebra - Fall 2015
HW #9
Please give
Section
Section
Section
complete, well-written solutions to the following exercises from Strang.
4.4: 22, 24
5.1: 1, 3, 18
5.2: 1, 13, 15, 16, 23
1
Linear Algebra - Fall 2015
HW #4
Please give
Section
Section
Section
complete, well-written solutions to the following exercises from Strang.
2.7: 13, 36
3.1: 18, 23, 30, 32
3.2: 18, 24, 36, 37
1
Linear Algebra - Fall 2015
HW #3
Please give
Section
Section
Section
complete, well-written solutions to the following exercises from Strang.
2.5: 15, 29, 30, 31,40
2.6: 5, 13, 14
2.7: 20, 37
1
Linear Algebra - Fall 2015
HW #6
Please give complete, well-written solutions to the following exercises from Strang.
Section 3.5: 2, 10, 16, 20, 24, 26, 41
Section 3.6: 2, 11, 24
1
Linear Algebra - Fall 2015
HW #2
Please give
Section
Section
Section
complete, well-written solutions to the following exercises from Strang.
2.2: 11, 14, 21, 26
2.3: 16, 23, 29
2.4: 16, 20, 34
1
Homework 5 - Solution
Due: Tuesday, October 17, 2017, 10:00PM
30 points total
1. (3 points for correctness) What linear combination of (1,2, 1) and (1,0,1) is closest to =
(2,2, 1)?
This is least squa