Linear Algebra, 2020a

182 Chapter Three. Maps Between Spaces
1.32 Suppose that f: V → W is an isomorphism. Prove that the set {⃗v 1 ,...,⃗v k } ⊆ V
is linearly dependent if and only if the set of images {f(⃗v 1 ),...,f(⃗v k )} ⊆ W is
linearly dependent.
̌ 1.33 Show that each type of map from Example 1.8 is an automorphism.
(a) Dilation d s by a nonzero scalar s.
(b) Rotation t θ through an angle θ.
(c) Reflection f l over a line through the origin.
Hint. For the second and third items, polar coordinates are useful.
1.34 Produce an automorphism of P 2 other than the identity map, and other than a
shift map p(x) ↦→ p(x − k).
1.35 (a) Show that a function f: R 1 → R 1 is an automorphism if and only if it has
the form x ↦→ kx for some k ≠ 0.
(b) Let f be an automorphism of R 1 such that f(3) =7. Find f(−2).
(c) Show that a function f: R 2 → R 2 is an automorphism if and only if it has the
form ( x
↦→
y)
( ) ax + by
cx + dy
for some a, b, c, d ∈ R with ad − bc ≠ 0. Hint. Exercises in prior subsections
have shown that ( ( b a
is not a multiple of
d)
c)
if and only if ad − bc ≠ 0.
(d) Let f be an automorphism of R 2 with
( ( ( 1 2 1
f( )= and f( )=
3)
−1)
4)
( 0
1)
.
Find
( 0
f( ).
−1)
1.36 Refer to Lemma 1.10 and Lemma 1.11. Find two more things preserved by
isomorphism.
1.37 We show that isomorphisms can be tailored to fit in that, sometimes, given
vectors in the domain and in the range we can produce an isomorphism associating
those vectors.
(a) Let B = 〈⃗β 1 , ⃗β 2 , ⃗β 3 〉 be a basis for P 2 so that any ⃗p ∈ P 2 has a unique
representation as ⃗p = c 1
⃗β 1 + c 2
⃗β 2 + c 3
⃗β 3 , which we denote in this way.
⎛ ⎞
c 1
Rep B (⃗p) = ⎝c 2
⎠
c 3
Show that the Rep B (·) operation is a function from P 2 to R 3 (this entails showing
that with every domain vector ⃗v ∈ P 2 there is an associated image vector in R 3 ,
and further, that with every domain vector ⃗v ∈ P 2 there is at most one associated
image vector).
(b) Show that this Rep B (·) function is one-to-one and onto.
(c) Show that it preserves structure.