A First Course in Linear Algebra, 2017a

562 Selected Exercise Answers
= [ ⃗ b1 ··· ⃗ bn
]
⃗ei
= ⃗ b i
Thus T⃗a i = [ ][ ]
⃗ b1 ··· ⃗ −1⃗ai
bn ⃗a1 ··· ⃗a n = A⃗a i .If⃗x is arbitrary, then since the matrix [ ]
⃗a 1 ··· ⃗a n
is invertible, there exists a unique⃗y such that [ ]
⃗a 1 ··· ⃗a n ⃗y =⃗x Hence
T⃗x = T
( n∑
i=1y i ⃗a i
)
=
n
∑
i=1
y i T⃗a i =
n
∑
i=1
y i A⃗a i = A
( n∑
i=1y i ⃗a i
)
= A⃗x
5.2.3 ⎡
⎣
5 1 5
1 1 3
3 5 −2
⎤⎡
⎦⎣
3 2 1
2 2 1
4 1 1
⎤
⎡
⎦ = ⎣
37 17 11
17 7 5
11 14 6
⎤
⎦
5.2.4 ⎡
⎣
1 2 6
3 4 1
1 1 −1
⎤⎡
⎦⎣
6 3 1
5 3 1
6 2 1
⎤
⎡
⎦ = ⎣
52 21 9
44 23 8
5 4 1
⎤
⎦
5.2.5 ⎡
⎣
−3 1 5
1 3 3
3 −3 −3
⎤⎡
⎦⎣
2 2 1
1 2 1
4 1 1
⎤
⎡
⎦ = ⎣
15 1 3
17 11 7
−9 −3 −3
⎤
⎦
5.2.6 ⎡
5.2.7 ⎡
⎣
⎣
3 1 1
3 2 3
3 3 −1
5 3 2
2 3 5
5 5 −2
⎤⎡
⎦⎣
⎤⎡
⎦⎣
6 2 1
5 2 1
6 1 1
11 4 1
10 4 1
12 3 1
⎤
⎡
⎦ = ⎣
⎤
⎡
⎦ = ⎣
29 9 5
46 13 8
27 11 5
⎤
⎦
109 38 10
112 35 10
81 34 8
5.2.11 Recall that proj ⃗u (⃗v)= ⃗v•⃗u ⃗u andsothedesiredmatrixhasi th column equal to proj
‖⃗u‖ 2 ⃗u (⃗e i ). Therefore,
the matrix desired is
⎡
⎤
1 −2 3
1
⎣ −2 4 −6 ⎦
14
3 −6 9
⎤
⎦
5.2.12
⎡
1
⎣
35
1 5 3
5 25 15
3 15 9
⎤
⎦