Linear Algebra, 2020a

220 Chapter Three. Maps Between Spaces
⎛ ⎞
⎛ ⎞
1 ( ) 0
(a) ⎝2⎠
0
(b) (c) ⎝0⎠
−1
1
0
1.15 Perform, if possible, each matrix-vector multiplication.
( ) ( ) 2 1 4 1 1 0
(a)
(b)
3 −1/2)( ⎛ ⎞
⎝1
⎠ (c)
2 −2 1 0
3
1
( 1 1
−2 1
) ⎛ ⎞
⎝1
3⎠
1
1.16 This matrix equation expresses a linear system. Solve it.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
2 1 1 x 8
⎝0 1 3⎠
⎝y⎠ = ⎝4⎠
1 −1 2 z 4
̌ 1.17 For a homomorphism from P 2 to P 3 that sends
1 ↦→ 1 + x, x ↦→ 1 + 2x, and x 2 ↦→ x − x 3
where does 1 − 3x + 2x 2 go?
1.18 Let h: R 2 → M 2×2 be the linear transformation with this action.
( ( ( ) ( )
1 1 2 0 0 −1
↦→
↦→
0)
0 1)
1 1 0
What is its effect on the general vector with entries x and y?
̌ 1.19 Assume that h: R 2 → R 3 is determined by this action.
⎛ ⎞
⎛ ⎞
( 2 ( 0
1
↦→ ⎝2⎠
0
↦→ ⎝ 1 ⎠
0)
1)
0
−1
Using the standard bases, find
(a) the matrix representing this map;
(b) a general formula for h(⃗v).
1.20 Represent the homomorphism h: R 3 → R 2 given by this formula and with
respect to these bases.
⎛ ⎞
x
⎝y⎠ ↦→
z
( ) x + y
x + z
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 1 (
B = 〈 ⎝1⎠ , ⎝1⎠ , ⎝
1
0⎠〉 D = 〈
0)
1 0 0
( 0
, 〉
2)
̌ 1.21 Let d/dx: P 3 → P 3 be the derivative transformation.
(a) Represent d/dx with respect to B, B where B = 〈1, x, x 2 ,x 3 〉.
(b) Represent d/dx with respect to B, D where D = 〈1, 2x, 3x 2 ,4x 3 〉.
̌ 1.22 Represent each linear map with respect to each pair of bases.
(a) d/dx: P n → P n with respect to B, B where B = 〈1,x,...,x n 〉, given by
a 0 + a 1 x + a 2 x 2 + ···+ a n x n ↦→ a 1 + 2a 2 x + ···+ na n x n−1
(b) ∫ : P n → P n+1 with respect to B n ,B n+1 where B i = 〈1,x,...,x i 〉, given by
a 0 + a 1 x + a 2 x 2 + ···+ a n x n ↦→ a 0 x + a 1
2 x2 + ···+ a n
n + 1 xn+1
(c) ∫ 1
0 : P n → R with respect to B, E 1 where B = 〈1,x,...,x n 〉 and E 1 = 〈1〉, given
by
a 0 + a 1 x + a 2 x 2 + ···+ a n x n ↦→ a 0 + a 1
2 + ···+ a n
n + 1