Linear Algebra, 2020a

Section III. Computing Linear Maps 221
(d) eval 3 : 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 · 3 + a 2 · 3 2 + ···+ a n · 3 n
(e) slide −1 : 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 0 + a 1 · (x + 1)+···+ a n · (x + 1) n
1.23 Represent the identity map on any nontrivial space with respect to B, B, where
B is any basis.
1.24 Represent, with respect to the natural basis, the transpose transformation on
the space M 2×2 of 2×2 matrices.
1.25 Assume that B = 〈⃗β 1 , ⃗β 2 , ⃗β 3 , ⃗β 4 〉 is a basis for a vector space. Represent with
respect to B, B the transformation that is determined by each.
(a) ⃗β 1 ↦→ ⃗β 2 , ⃗β 2 ↦→ ⃗β 3 , ⃗β 3 ↦→ ⃗β 4 , ⃗β 4 ↦→ ⃗0
(b) ⃗β 1 ↦→ ⃗β 2 , ⃗β 2 ↦→ ⃗0, ⃗β 3 ↦→ ⃗β 4 , ⃗β 4 ↦→ ⃗0
(c) ⃗β 1 ↦→ ⃗β 2 , ⃗β 2 ↦→ ⃗β 3 , ⃗β 3 ↦→ ⃗0, ⃗β 4 ↦→ ⃗0
1.26 Example 1.10 shows how to represent the rotation transformation of the plane
with respect to the standard basis. Express these other transformations also with
respect to the standard basis.
(a) the dilation map d s , which multiplies all vectors by the same scalar s
(b) the reflection map f l , which reflects all all vectors across a line l through the
origin
̌ 1.27 Consider a linear transformation of R 2 determined by these two.
( ( ) ( ( )
1 2 1 −1
↦→
↦→
1)
0 0)
0
(a) Represent this transformation with respect to the standard bases.
(b) Where does the transformation send this vector?
( 0
5)
(c) Represent this transformation with respect to these bases.
( ) ( ( ( )
1 1 2 −1
B = 〈 , 〉 D = 〈 , 〉
−1 1)
2)
1
(d) Using B from the prior item, represent the transformation with respect to
B, B.
1.28 Suppose that h: V → W is one-to-one so that by Theorem 2.20, for any basis
B = 〈⃗β 1 ,...,⃗β n 〉⊂V the image h(B) =〈h(⃗β 1 ),...,h(⃗β n )〉 is a basis for h(V). (If
h is onto then h(V) =W.)
(a) Represent the map h with respect to 〈B, h(B)〉.
(b) For a member ⃗v of the domain, where the representation of ⃗v has components
c 1 , ..., c n , represent the image vector h(⃗v) with respect to the image basis h(B).
1.29 Give a formula for the product of a matrix and ⃗e i , the column vector that is
all zeroes except for a single one in the i-th position.
̌ 1.30 For each vector space of functions of one real variable, represent the derivative
transformation with respect to B, B.
(a) {a cos x + b sin x | a, b ∈ R}, B = 〈cos x, sin x〉