Linear Algebra

164 Chapter Three. Maps Between Spaces(a) Example 1.1 (b) Example 1.2̌ 1.12 For the map f: P 1 → R 2 given by( )a + bx ↦−→f a − bbFind the image of each of these elements of the domain.(a) 3 − 2x (b) 2 + 2x (c) xShow that this map is an isomorphism.1.13 Show that the natural map f 1 from Example 1.5 is an isomorphism.̌ 1.14 Decide whether each map is an isomorphism (if it is an isomorphism then proveit and if it isn’t then state a condition that it fails to satisfy).(a) f: M 2×2 → R given by( ) a b↦→ ad − bcc d(b) f: M 2×2 → R 4 given by⎛⎞a + b + c + d( ) a b↦→ ⎜ a + b + c⎟c d ⎝ a + b ⎠a(c) f: M 2×2 → P 3 given by( ) a b↦→ c + (d + c)x + (b + a)x 2 + ax 3c d(d) f: M 2×2 → P 3 given by( ) a b↦→ c + (d + c)x + (b + a + 1)x 2 + ax 3c d1.15 Show that the map f: R 1 → R 1 given by f(x) = x 3 is one-to-one and onto. Is itan isomorphism?̌ 1.16 Refer to Example 1.1. Produce two more isomorphisms (of course, you mustalso verify that they satisfy the conditions in the definition of isomorphism).1.17 Refer to Example 1.2. Produce two more isomorphisms (and verify that theysatisfy the conditions).̌ 1.18 Show that, although R 2 is not itself a subspace of R 3 , it is isomorphic to thexy-plane subspace of R 3 .1.19 Find two isomorphisms between R 16 and M 4×4 .̌ 1.20 For what k is M m×n isomorphic to R k ?1.21 For what k is P k isomorphic to R n ?1.22 Prove that the map in Example 1.8, from P 5 to P 5 given by p(x) ↦→ p(x − 1),is a vector space isomorphism.1.23 Why, in Lemma 1.9, must there be a ⃗v ∈ V? That is, why must V be nonempty?1.24 Are any two trivial spaces isomorphic?1.25 In the proof of Lemma 1.10, what about the zero-summands case (that is, if nis zero)?1.26 Show that any isomorphism f: P 0 → R 1 has the form a ↦→ ka for some nonzeroreal number k.̌ 1.27 These prove that isomorphism is an equivalence relation.(a) Show that the identity map id: V → V is an isomorphism. Thus, any vectorspace is isomorphic to itself.