Linear Algebra

168 Chapter Three. Maps Between Spacesthe meaning of same-ness that was of interest there). We showed that is anequivalence relation and so the collection of matrices is partitioned into classes,where all the matrices that are row equivalent fall together into a single class.Then, for insight into which matrices are in each class, we gave representativesfor the classes, the reduced echelon form matrices.In this section, except that the appropriate notion of same-ness here is vectorspace isomorphism, we have followed much the same outline. First we definedisomorphism, saw some examples, and established some properties. Then weshowed that it is an equivalence relation, and now we have a set of class representatives,the real vector spaces R 1 , R 2 , etc.⋆ R 0 ⋆ R 3All finite dimensionalvector spaces: ⋆ R 2. . .⋆ R 1One representativeper classAs before, the list of representatives helps us to understand the partition. It issimply a classification of spaces by dimension.In the second chapter, with the definition of vector spaces, we seemed tohave opened up our studies to many examples of new structures besides thefamiliar R n ’s. We now know that isn’t the case. Any finite-dimensional vectorspace is actually “the same” as a real space. We are thus considering exactlythe structures that we need to consider.The rest of the chapter fills out the work in this section. In particular,in the next section we will consider maps that preserve structure, but are notnecessarily correspondences.Exerciseš 2.8 Decide if the spaces are isomorphic.(a) R 2 , R 4 (b) P 5, R 5 (c) M 2×3, R 6 (d) P 5, M 2×3 (e) M 2×k , C ǩ 2.9 Consider the isomorphism Rep B (·): P 1 → R 2 where B = 〈1, 1 + x〉. Find theimage of each of these elements of the domain.(a) 3 − 2x; (b) 2 + 2x; (c) x̌ 2.10 Show that if m ≠ n then R m ≁ = R n .̌ 2.11 Is M m×n∼ = Mn×m?̌ 2.12 Are any two planes through the origin in R 3 isomorphic?2.13 Find a set of equivalence class representatives other than the set of R n ’s.2.14 True or false: between any n-dimensional space and R n there is exactly oneisomorphism.2.15 Can a vector space be isomorphic to one of its (proper) subspaces?̌ 2.16 This subsection shows that for any isomorphism, the inverse map is also an isomorphism.This subsection also shows that for a fixed basis B of an n-dimensionalvector space V , the map Rep B : V → R n is an isomorphism. Find the inverse ofthis map.̌ 2.17 Prove these facts about matrices.(a) The row space of a matrix is isomorphic to the column space of its transpose.