Linear Algebra

172 Chapter Three. Maps Between Spaces(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 2.10 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 2.11 Show that if m ≠ n then R m ̸∼ = R n .̌ 2.12 Is M m×n∼ = Mn×m ?̌ 2.13 Are any two planes through the origin in R 3 isomorphic?2.14 Find a set of equivalence class representatives other than the set of R n ’s.2.15 True or false: between any n-dimensional space and R n there is exactly oneisomorphism.2.16 Can a vector space be isomorphic to one of its (proper) subspaces?̌ 2.17 This subsection shows that for any isomorphism, the inverse map is also anisomorphism. 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.̌ 2.18 Prove these facts about matrices.(a) The row space of a matrix is isomorphic to the column space of its transpose.(b) The row space of a matrix is isomorphic to its column space.2.19 Show that the function from Theorem 2.3 is well-defined.2.20 Is the proof of Theorem 2.3 valid when n = 0?2.21 For each, decide if it is a set of isomorphism class representatives.(a) {C k ∣ ∣ k ∈ N}(b) {P k∣ ∣ k ∈ {−1, 0, 1, . . .}}(c) {M m×n∣ ∣ m, n ∈ N}2.22 Let f be a correspondence between vector spaces V and W (that is, a map thatis one-to-one and onto). Show that the spaces V and W are isomorphic via f if andonly if there are bases B ⊂ V and D ⊂ W such that corresponding vectors have thesame coordinates: Rep B (⃗v) = Rep D (f(⃗v)).2.23 Consider the isomorphism Rep B : P 3 → R 4 .(a) Vectors in a real space are orthogonal if and only if their dot product is zero.Give a definition of orthogonality for polynomials.(b) The derivative of a member of P 3 is in P 3 . Give a definition of the derivativeof a vector in R 4 .̌ 2.24 Does every correspondence between bases, when extended to the spaces, givean isomorphism?2.25 (Requires the subsection on Combining Subspaces, which is optional.) Supposethat V = V 1 ⊕ V 2 and that V is isomorphic to the space U under the map f.Show that U = f(V 1 ) ⊕ f(U 2 ).2.26 Show that this is not a well-defined function from the rational numbers to theintegers: with each fraction, associate the value of its numerator.