Linear Algebra

166 Chapter Three. Maps Between Spaces(b) Show that this Rep B (·) function is one-to-one and onto.(c) Show that it preserves structure.(d) Produce an isomorphism from P 2 to R 3 that fits these specifications.⎛ ⎞1x + x 2 ↦→ ⎝0⎠ and⎛ ⎞01 − x ↦→ ⎝1⎠001.35 Prove that a space is n-dimensional if and only if it is isomorphic to R n . Hint.Fix a basis B for the space and consider the map sending a vector over to itsrepresentation with respect to B.1.36 (Requires the subsection on Combining Subspaces, which is optional.) LetU and W be vector spaces. Define a new vector space, consisting of the setU × W = {(⃗u, ⃗w) ∣ ⃗u ∈ U and ⃗w ∈ W } along with these operations.(⃗u 1 , ⃗w 1 ) + (⃗u 2 , ⃗w 2 ) = (⃗u 1 + ⃗u 2 , ⃗w 1 + ⃗w 2 ) and r · (⃗u, ⃗w) = (r⃗u, r⃗w)This is a vector space, the external direct sum of U and W.(a) Check that it is a vector space.(b) Find a basis for, and the dimension of, the external direct sum P 2 × R 2 .(c) What is the relationship among dim(U), dim(W), and dim(U × W)?(d) Suppose that U and W are subspaces of a vector space V such that V = U ⊕ W(in this case we say that V is the internal direct sum of U and W). Show thatthe map f: U × W → V given byf(⃗u, ⃗w) ↦−→ ⃗u + ⃗wis an isomorphism. Thus if the internal direct sum is defined then the internaland external direct sums are isomorphic.I.2 Dimension Characterizes IsomorphismIn the prior subsection, after stating the definition of an isomorphism, we gavesome results supporting the intuition that such a map describes spaces as “thesame.” Here we will develop this intuition. When two spaces that are isomorphicare not equal, we think of them as almost equal, as equivalent. We shall showthat the relationship ‘is isomorphic to’ is an equivalence relation. ∗2.1 Lemma The inverse of an isomorphism is also an isomorphism.Proof Suppose that V is isomorphic to W via f: V → W. Because an isomorphismis a correspondence, f has an inverse function f −1 : W → V that is also acorrespondence. †To finish we will show that because f preserves linear combinations, so also∗ More information on equivalence relations and equivalence classes is in the appendix.† More information on inverse functions is in the appendix.