Linear Algebra

Section I. Isomorphisms 167
(d) f : M2×2 →P3 given by
� �
a b
c d
↦→ c +(d + c)x +(b + a +1)x 2 + ax 3
1.14 Show that the map f : R 1 → R 1 given by f(x) =x 3 is one-to-one and onto.
Is it an isomorphism?
� 1.15 Refer to Example 1.1. Produce two more isomorphisms (of course, that they
satisfy the conditions in the definition of isomorphism must be verified).
1.16 Refer to Example 1.2. Produce two more isomorphisms (and verify that they
satisfy the conditions).
� 1.17 Show that, although R 2 is not itself a subspace of R 3 , it is isomorphic to the
xy-plane subspace of R 3 .
1.18 Find two isomorphisms between R 16 and M4×4.
� 1.19 For what k is Mm×n isomorphic to R k ?
1.20 For what k is Pk isomorphic to R n ?
1.21 Prove that the map in Example 1.6, fromP5 to P5 given by p(x) ↦→ p(x − 1),
is a vector space isomorphism.
1.22 Why, in Lemma 1.8, must there be a �v ∈ V ? That is, why must V be
nonempty?
1.23 Are any two trivial spaces isomorphic?
1.24 In the proof of Lemma 1.9, what about the zero-summands case (that is, if n
is zero)?
1.25 Show that any isomorphism f : P0 → R 1 has the form a ↦→ ka for some nonzero
real number k.
� 1.26 These prove that isomorphism is an equivalence relation.
(a) Show that the identity map id: V → V is an isomorphism. Thus, any vector
space is isomorphic to itself.
(b) Show that if f : V → W is an isomorphism then so is its inverse f −1 : W → V .
Thus, if V is isomorphic to W then also W is isomorphic to V .
(c) Show that a composition of isomorphisms is an isomorphism: if f : V → W is
an isomorphism and g : W → U is an isomorphism then so also is g ◦ f : V → U.
Thus, if V is isomorphic to W and W is isomorphic to U, thenalsoV is isomorphic
to U.
1.27 Suppose that f : V → W preserves structure. Show that f is one-to-one if and
only if the unique member of V mapped by f to �0W is �0V .
1.28 Suppose that f : V → W is an isomorphism. Prove that the set {�v1,...,�vk} ⊆
V is linearly dependent if and only if the set of images {f(�v1),...,f(�vk)} ⊆W is
linearly dependent.
� 1.29 Show that each type of map from Example 1.7 is an automorphism.
(a) Dilation ds by a nonzero scalar s.
(b) Rotation tθ throughanangleθ.
(c) Reflection fℓ over a line through the origin.
Hint. For the second and third items, polar coordinates are useful.
1.30 Produce an automorphism of P2 other than the identity map, and other than
ashiftmapp(x) ↦→ p(x − k).
1.31 (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).