A First Course in Linear Algebra, 2017a

528 Vector Spaces
⎡
(a) T ⎣
⎡
(b) T ⎣
⎡
(c) T ⎣
⎡
(d) T ⎣
x
y
z
x
y
z
x
y
z
x
y
z
⎤
⎦ =
⎤
[ x + 2y + 3z + 1
2y − 3x + z
⎦ =
[
x + 2y 2 + 3z
2y + 3x + z
⎤
⎦ =
⎤
⎦ =
]
[
sinx + 2y + 3z
2y + 3x + z
[ x + 2y + 3z
2y + 3x − lnz
]
]
]
Exercise 9.9.10 Suppose
[
A1 ··· A n
] −1
exists where each A j ∈ R n and let vectors {B 1 ,···,B n } in R m be given. Show that there always exists a
linear transformation T such that T(A i )=B i .
Exercise 9.9.11 Find the matrix for T (⃗w)=proj ⃗v (⃗w) where ⃗v = [ 1 −2 3 ] T .
Exercise 9.9.12 Find the matrix for T (⃗w)=proj ⃗v (⃗w) where ⃗v = [ 1 5 3 ] T .
Exercise 9.9.13 Find the matrix for T (⃗w)=proj ⃗v (⃗w) where ⃗v = [ 1 0 3 ] T .
{[ ] [ ]}
[ ]
2 3
Exercise 9.9.14 Let B = , be a basis of R
−1 2
2 5
and let ⃗x = be a vector in R
−7
2 .Find
C B (⃗x).
⎧⎡
⎤ ⎡ ⎤ ⎡ ⎤⎫
⎡ ⎤
⎨ 1 2 −1 ⎬
5
Exercise 9.9.15 Let B = ⎣ −1 ⎦, ⎣ 1 ⎦, ⎣ 0 ⎦
⎩
⎭ be a basis of R3 and let⃗x = ⎣ −1 ⎦ be a vector
2 2 2
4
in R 2 .FindC B (⃗x).
([ ]) [ ]
a a + b
Exercise 9.9.16 Let T : R 2 ↦→ R 2 be a linear transformation defined by T = .
b a − b
Consider the two bases
{[ ] [ ]}
1 −1
B 1 = {⃗v 1 ,⃗v 2 } = ,
0 1
and
{[
1
B 2 =
1
] [
,
1
−1
Find the matrix M B2 ,B 1
of T with respect to the bases B 1 and B 2 .
]}