Linear Algebra

Section I. Isomorphisms 165(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 isan 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, then also V is isomorphicto U.1.28 Suppose that f: V → W preserves structure. Show that f is one-to-one if andonly if the unique member of V mapped by f to ⃗0 W is ⃗0 V .1.29 Suppose that f: V → W is an isomorphism. Prove that the set {⃗v 1 , . . . ,⃗v k } ⊆ Vis linearly dependent if and only if the set of images {f(⃗v 1 ), . . . , f(⃗v k )} ⊆ W islinearly dependent.̌ 1.30 Show that each type of map from Example 1.7 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.1.31 Produce an automorphism of P 2 other than the identity map, and other than ashift map p(x) ↦→ p(x − k).1.32 (a) Show that a function f: R 1 → R 1 is an automorphism if and only if it hasthe 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 theform ( ( )x ax + by↦→y)cx + dyfor some a, b, c, d ∈ R with ad − bc ≠ 0. Hint. Exercises in prior subsectionshave shown that ( ( b ais not a multiple ofd)c)if and only if ad − bc ≠ 0.(d) Let f be an automorphism of R 2 with( ( ) ( ( 1 21 0f( ) = and f( ) = .3)−14)1)Find( ) 0f( ).−11.33 Refer to Lemma 1.9 and Lemma 1.10. Find two more things preserved byisomorphism.1.34 We show that isomorphisms can be tailored to fit in that, sometimes, givenvectors in the domain and in the range we can produce an isomorphism associatingthose vectors.(a) Let B = 〈⃗β 1 , ⃗β 2 , ⃗β 3 〉 be a basis for P 2 so that any ⃗p ∈ P 2 has a uniquerepresentation as ⃗p = c 1⃗β 1 + c 2⃗β 2 + c 3⃗β 3 , which we denote in this way.⎛ ⎞c 1Rep B (⃗p) = ⎝c 2⎠c 3Show that the Rep B (·) operation is a function from P 2 to R 3 (this entails showingthat 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 associatedimage vector).