$ \begin{array}{l} \textbf{Exercises 3E} \\ 1. \text{ Suppose } T \text{ is a function from } V \text{ to } W.\\ \text{ The graph of function } T \text{ is the subset of } V \times W \\ \text{ defined by} \\ \text{graph of } T = {(v,Tv)\in V x W:v \in V}\\\\ \text{ Prove that } T \text{ is a linear map if and only if the graph of } T \\\text{ is a subspace of } V \times W. \\ \text{ Formally, a function } T \text{ from } V \text{ to } W \text{ is a subset } T \text{ of } V \times W \text{ such that for each } v \in V, \\ \text{ there exists exactly one element } (v, w) \in T. \\\\\text{ In other words, formally a function is what is called above its graph.} \\\\ \text{ We do not usually think of functions in this formal manner.}\\\text{ However, if we do become formal, then this exercise could be rephrased as follows:} \\ \text{ Prove that a function } T \text{ from } V \text{ to } W \text{ is a linear map if and only if } T \text{ is a subspace of } V \times W. \\\\ 2. \text{ Suppose that } V_1, \ldots, V_m \text{ are vector spaces such that } V_1 \times \cdots \times V_m \text{ is finite-dimensional.}\\\text{ Prove that } \\V_k \text{ is finite-dimensional for each } k = 1, \ldots, m. \\\\ 3. \text{ Suppose } V_1, \ldots, V_m \text{ are vector spaces. Prove that } \\L(V_1 \times \cdots \times V_m, W) \text{ and } L(V_1, W) \times \cdots \times L(V_m, W) \text{ are isomorphic vector spaces.} \\\\ 4. \text{ Suppose } W_1, \ldots, W_m \text{ are vector spaces. Prove that }\\ L(V, W_1 \times \cdots \times W_m) \text{ and } L(V, W_1) \times \cdots \times L(V, W_m) \text{ are isomorphic vector spaces.} \\\\ 5. \text{ For } m \text{ a positive integer, define } V_m \text{ by } V_m = V \times \cdots \times V \text{ (} m \text{ times)}. \\\text{ Prove that } V_m \text{ and } L(F^m, V) \text{ are isomorphic vector spaces.} \\\\ 6. \text{ Suppose that } v, x \text{ are vectors in } V \text{ and that } U, W \text{ are subspaces of } V \text{ such that } v + U = x + W.\\ \text{ Prove that } U = W. \\ 7. \text{ Let } U = \{(x, y, z) \in \mathbb{R}^3 : 2x + 3y + 5z = 0\}. \text{ Suppose } A \subseteq \mathbb{R}^3.\\ \text{ Prove that } A \text{ is a translate of } U \text{ if and only if there exists } c \in \mathbb{R} \text{ such that } \\ A = \{(x, y, z) \in \mathbb{R}^3 : 2x + 3y + 5z = c\}. \\\\ 8. (a) \text{ Suppose } T \in L(V, W) \text{ and } c \in W. \text{ Prove that } \{x \in V : T x = c\} \\\text{ is either the empty set or is a translate of null } T. \\\\ (b) \text{ Explain why the set of solutions to a system of linear equations such as 3.27}\\\text{ is either the empty set or is a translate of some subspace of } F^n. \\\\ 9. \text{ Prove that a nonempty subset } A \text{ of } V \text{ is a translate of some subspace of } V \\\text{ if and only if } \lambda v + (1 - \lambda) w \in A \text{ for all } v, w \in A \text{ and all } \lambda \in F. \\ 10. \text{ Suppose } A_1 = v + U_1 \text{ and } A_2 = w + U_2 \text{ for some } v, w \in V \text{ and some subspaces } U_1, U_2 \text{ of } V.\\ \text{ Prove that the intersection } A_1 \cap A_2 \text{ is either a translate of some subspace of } V \text{ or is the empty set.} \\ \text{ graph of } T = \{(v, T v) \in V \times W : v \in V\}. \end{array} $ --- $\begin{array}{l} 104 \text{ Chapter 3 Linear Maps} \\ 11. \text{ Suppose } U = \{(x_1, x_2, \ldots) \in F^\infty : x_k \neq 0 \\\text{ for only finitely many } k\}. \\ (a) \text{ Show that } U \text{ is a subspace of } F^\infty. \\ (b) \text{ Prove that } F^\infty/U \text{ is infinite-dimensional.} \\\\ 12. \text{ Suppose } v_1, \ldots, v_m \in V. \text{ Let } A = \{\lambda_1 v_1 + \cdots + \lambda_m v_m : \lambda_1, \ldots, \lambda_m \in F \text{ and } \lambda_1 + \cdots + \lambda_m = 1\}. \\ (a) \text{ Prove that } A \text{ is a translate of some subspace of } V. \\ (b) \text{ Prove that if } B \text{ is a translate of some subspace of } V \text{ and } \{v_1, \ldots, v_m\} \subseteq B, \\ \text{ then } A \subseteq B. \\ (c) \text{ Prove that } A \text{ is a translate of some subspace of } V \text{ of dimension less than } m. \\\\ 13. \text{ Suppose } U \text{ is a subspace of } V \text{ such that } V/U \\\text{ is finite-dimensional.\\ Prove that } V \text{ is isomorphic to } U \times (V/U). \\\\ 14. \text{ Suppose } U \text{ and } W \text{ are subspaces of } V \text{ and } V = U \oplus W. \text{ Suppose } w_1, \ldots, w_m \text{ is a basis of } W. \\\\ \text{ Prove that } w_1 + U, \ldots, w_m + U \text{ is a basis of } V/U. \\\\ 15. \text{ Suppose } U \text{ is a subspace of } V \text{ and } v_1 + U, \ldots, v_m + U \text{ is a basis of } V/U \text{ and } u_1, \ldots, u_n \\\text{ is a basis of } U. \\ \text{ Prove that } v_1, \ldots, v_m, u_1, \ldots, u_n \text{ is a basis of } V. \\\\ 16. \text{ Suppose } \phi \in L(V, F) \text{ and } \phi \neq 0. \text{ Prove that } \text{dim } V/(\text{null } \phi) = 1. \\\\ 17. \text{ Suppose } U \text{ is a subspace of } V \text{ such that } \text{dim } V/U = 1. \\\text{ Prove that there exists } \phi \in L(V, F) \text{ such that } \text{null } \phi = U. \\\\ 18. \text{ Suppose that } U \text{ is a subspace of } V \text{ such that } V/U \text{ is finite-dimensional.} \\ (a) \text{ Show that if } W \text{ is a finite-dimensional subspace of } V \text{ and } V = U + W, \text{ then } \text{dim } W \geq \text{dim } V/U. \\ (b) \text{ Prove that there exists a finite-dimensional subspace } W\\ \text{ of } V \text{ such that } \text{dim } W = \text{dim } V/U \text{ and } V = U \oplus W. \\\\ 19. \text{ Suppose } T \in L(V, W) \text{ and } U \text{ is a subspace of } V.\\ \text{ Let } \pi \text{ denote the quotient map from } V \text{ onto } V/U. \\\text{ Prove that there exists } S \in L(V/U, W) \text{ such that } T = S \circ \pi \text{ if and only if } U \subseteq \text{null } T. \end{array} $ --- Exercises 3E 1 Suppose 𝑇 is a function from 𝑉 to 𝑊. The graph of 𝑇 is the subset of 𝑉× 𝑊 defined by Prove that 𝑇 is a linear map if and only if the graph of 𝑇 is a subspace of 𝑉×𝑊. Formally, a function 𝑇 from 𝑉 to 𝑊 is a subset 𝑇 of 𝑉× 𝑊 such that for each 𝑣 ∈ 𝑉, there exists exactly one element (𝑣, 𝑤) ∈ 𝑇. In other words, formally a function is what is called above its graph. We do not usually think of functions in this formal manner. However, if we do become formal, then this exercise could be rephrased as follows: Prove that a function 𝑇 from𝑉to𝑊isalinearmapifandonlyif 𝑇isasubspaceof 𝑉×𝑊. 2. 3E-2  Suppose that 𝑉1, ..., 𝑉𝑚 are vector spaces such that 𝑉1 × ⋯ × 𝑉𝑚 is finite- dimensional. Prove that 𝑉𝑘 is finite-dimensional for each 𝑘 = 1, ..., 𝑚. 3. 3E-3  Suppose𝑉1,...,𝑉𝑚arevectorspaces.ProvethatL(𝑉1×⋯×𝑉𝑚,𝑊)and L(𝑉1, 𝑊) × ⋯ × L(𝑉𝑚, 𝑊) are isomorphic vector spaces. 4. 3E-4  Suppose 𝑊1, ..., 𝑊𝑚 are vector spaces. Prove that L(𝑉, 𝑊1 × ⋯ × 𝑊𝑚) and L(𝑉, 𝑊1) × ⋯ × L(𝑉, 𝑊𝑚) are isomorphic vector spaces. 5. 3E-5  For 𝑚 a positive integer, define 𝑉𝑚 by 𝑉 𝑚 = 𝑉 ⏟× ⋯ × 𝑉 . 𝑚 times Prove that 𝑉𝑚 and L(𝐅𝑚, 𝑉) are isomorphic vector spaces. 6. 3E-6  Suppose that 𝑣, 𝑥 are vectors in 𝑉 and that 𝑈, 𝑊 are subspaces of 𝑉 such that 𝑣 + 𝑈 = 𝑥 + 𝑊. Prove that 𝑈 = 𝑊. 7. 3E-7  Let𝑈={(𝑥,𝑦,𝑧)∈𝐑3 ∶2𝑥+3𝑦+5𝑧=0}.Suppose𝐴⊆𝐑3.Provethat 𝐴 is a translate of 𝑈 if and only if there exists 𝑐 ∈ 𝐑 such that 𝐴 = {(𝑥,𝑦,𝑧) ∈ 𝐑3 ∶2𝑥+3𝑦+5𝑧 = 𝑐}. 8. 3E-8  (a) Suppose𝑇 ∈ L(𝑉,𝑊)and𝑐 ∈ 𝑊. Provethat{𝑥 ∈ 𝑉 ∶𝑇𝑥 = 𝑐}is either the empty set or is a translate of null 𝑇. (b) Explainwhythesetofsolutionstoasystemoflinearequationssuchas 3.27 is either the empty set or is a translate of some subspace of 𝐅𝑛. 9. 3E-9  Prove that a nonempty subset 𝐴 of 𝑉 is a translate of some subspace of 𝑉 if and only if 𝜆𝑣 + (1 − 𝜆)𝑤 ∈ 𝐴 for all 𝑣, 𝑤 ∈ 𝐴 and all 𝜆 ∈ 𝐅. 10. 3E-10  Suppose𝐴1 = 𝑣+𝑈1 and𝐴2 = 𝑤+𝑈2 forsome𝑣,𝑤 ∈ 𝑉andsome subspaces 𝑈1,𝑈2 of 𝑉. Prove that the intersection 𝐴1 ∩ 𝐴2 is either a translate of some subspace of 𝑉 or is the empty set. graph of 𝑇 = {(𝑣, 𝑇𝑣) ∈ 𝑉 × 𝑊 ∶ 𝑣 ∈ 𝑉}. 104 Chapter 3 Linear Maps 11. 3E-11  Suppose𝑈={(𝑥1,𝑥2,...)∈𝐅∞ ∶𝑥𝑘 ≠0foronlyfinitelymany𝑘}. (a) Showthat𝑈isasubspaceof𝐅∞. (b) Prove that 𝐅∞/𝑈 is infinite-dimensional. 12. 3E-12  Suppose 𝑣1, ..., 𝑣𝑚 ∈ 𝑉. Let 𝐴={𝜆1𝑣1+⋯+𝜆𝑚𝑣𝑚 ∶𝜆1,...,𝜆𝑚 ∈𝐅and𝜆1+⋯+𝜆𝑚 =1}. 1. (a)  Provethat𝐴isatranslateofsomesubspaceof𝑉. 2. (b)  Provethatif𝐵isatranslateofsomesubspaceof𝑉and{𝑣1,...,𝑣𝑚}⊆𝐵, then 𝐴 ⊆ 𝐵. 3. (c)  Prove that 𝐴 is a translate of some subspace of 𝑉 of dimension less than 𝑚. 13. 3E-13  Suppose 𝑈 is a subspace of 𝑉 such that 𝑉/𝑈 is finite-dimensional. Prove that 𝑉 is isomorphic to 𝑈 × (𝑉/𝑈). 14. 3E-14  Suppose𝑈and𝑊aresubspacesof𝑉and𝑉=𝑈⊕𝑊.Suppose𝑤1,...,𝑤𝑚 isabasisof𝑊. Provethat𝑤1 +𝑈,...,𝑤𝑚 +𝑈isabasisof𝑉/𝑈. 15. 3E-15  Suppose𝑈isasubspaceof𝑉and𝑣1+𝑈,...,𝑣𝑚+𝑈isabasisof𝑉/𝑈and 𝑢1,...,𝑢𝑛 is a basis of 𝑈. Prove that 𝑣1,...,𝑣𝑚,𝑢1,...,𝑢𝑛 is a basis of 𝑉. 16. 3E-16  Suppose𝜑∈L(𝑉,𝐅)and𝜑≠0.Provethatdim𝑉/(null𝜑)=1. 17. 3E-17  Suppose 𝑈 is a subspace of 𝑉 such that dim 𝑉/𝑈 = 1. Prove that there exists 𝜑 ∈ L(𝑉,𝐅) such that null𝜑 = 𝑈. 18. 3E-18  Suppose that 𝑈 is a subspace of 𝑉 such that 𝑉/𝑈 is finite-dimensional. 1. (a)  Showthatif𝑊isafinite-dimensionalsubspaceof𝑉and𝑉=𝑈+𝑊, then dim 𝑊 ≥ dim 𝑉/𝑈. 2. (b)  Provethatthereexistsafinite-dimensionalsubspace𝑊of𝑉suchthat dim𝑊 =dim𝑉/𝑈and𝑉 =𝑈⊕𝑊. 19. 3E-19  Suppose 𝑇 ∈ L(𝑉, 𝑊) and 𝑈 is a subspace of 𝑉. Let 𝜋 denote the quotient map from 𝑉 onto 𝑉/𝑈. Prove that there exists 𝑆 ∈ L(𝑉/𝑈, 𝑊) such that 𝑇 = 𝑆 ∘ 𝜋 if and only if 𝑈 ⊆ null 𝑇.