$ \begin{array}{l} \textbf{Exercises 3E} \\ 1. \text{ Suppose } T \text{ is a function from } V \text{ to } W. \text{ The graph of } 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. 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 } \text{null } T. \\ (b) \text{ Explain why the set of solutions to a system of linear equations such as 3.27 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\}. 104 Chapter 3 Linear Maps 11 Suppose𝑈={(𝑥1,𝑥2,...)∈𝐅∞ ∶𝑥𝑘 ≠0foronlyfinitelymany𝑘}. (a) Showthat𝑈isasubspaceof𝐅∞. (b) Prove that 𝐅∞/𝑈 is infinite-dimensional. 9. 12 Suppose 𝑣1, ..., 𝑣𝑚 ∈ 𝑉. Let 𝐴={𝜆1𝑣1+⋯+𝜆𝑚𝑣𝑚 ∶𝜆1,...,𝜆𝑚 ∈𝐅and𝜆1+⋯+𝜆𝑚 =1}. (a) Provethat𝐴isatranslateofsomesubspaceof𝑉. (b) Provethatif𝐵isatranslateofsomesubspaceof𝑉and{𝑣1,...,𝑣𝑚}⊆𝐵, then 𝐴 ⊆ 𝐵. (c) Prove that 𝐴 is a translate of some subspace of 𝑉 of dimension less than 𝑚. 10. 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 Suppose𝑈and𝑊aresubspacesof𝑉and𝑉=𝑈⊕𝑊.Suppose𝑤1,...,𝑤𝑚 isabasisof𝑊. Provethat𝑤1 +𝑈,...,𝑤𝑚 +𝑈isabasisof𝑉/𝑈. 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 Suppose𝜑∈L(𝑉,𝐅)and𝜑≠0.Provethatdim𝑉/(null𝜑)=1. 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 Suppose that 𝑈 is a subspace of 𝑉 such that 𝑉/𝑈 is finite-dimensional. (a) Showthatif𝑊isafinite-dimensionalsubspaceof𝑉and𝑉=𝑈+𝑊, then dim 𝑊 ≥ dim 𝑉/𝑈. (b) Provethatthereexistsafinite-dimensionalsubspace𝑊of𝑉suchthat dim𝑊 =dim𝑉/𝑈and𝑉 =𝑈⊕𝑊. 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. 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} $ 1.$ \begin{array}{l} \text{To show that \( T \) is a linear map}\\\text{ if and only if the graph of \( T \) is a subspace of \( V \times W \),}\\\text{ let's prove both directions.}\\ \text{(a) If \( T \) is a linear map, then for any vectors \( (v_1, w_1) \) and \( (v_2, w_2) \)}\\\text{ in the graph of \( T \), we have \( T(v_1 + v_2) = T(v_1) + T(v_2) = w_1 + w_2 \),} and \( T(\lambda v_1) = \lambda T(v_1) = \lambda w_1 \) for any scalar \( \lambda \). This means that \( (v_1 + v_2, w_1 + w_2) \) and \( (\lambda v_1, \lambda w_1) \) are also in the graph of \( T \), satisfying the closure under vector addition and scalar multiplication. (b) Conversely, if the graph of \( T \) is a subspace of \( V \times W \), let's define \( T \) on any vector \( v \) in \( V \) as \( T(v) = w \), where \( (v, w) \) is the corresponding vector in the graph of \( T \). It's straightforward to show that \( T \) is linear. 2. To prove that \( V_k \) is finite-dimensional for each \( k = 1, ..., m \), we can use the fact that the product of finite-dimensional vector spaces is finite-dimensional. Since \( V_1 \times \ldots \times V_m \) is finite-dimensional, each factor \( V_k \) must also be finite-dimensional. 3. To prove the isomorphism between \( L(V_1 \times \ldots \times V_m, W) \) and \( L(V_1, W) \times \ldots \times L(V_m, W) \), we can define a linear map \( \Phi: L(V_1 \times \ldots \times V_m, W) \rightarrow L(V_1, W) \times \ldots \times L(V_m, W) \) by \( \Phi(T) = (T_1, \ldots, T_m) \), where \( T_k \) is the projection of \( T \) onto the \( k \)-th component. This map is an isomorphism. 4. Similarly, to prove the isomorphism between \( L(V, W_1 \times \ldots \times W_m) \) and \( L(V, W_1) \times \ldots \times L(V, W_m) \), we can define a linear map \( \Psi: L(V, W_1 \times \ldots \times W_m) \rightarrow L(V, W_1) \times \ldots \times L(V, W_m) \) by \( \Psi(T) = (T_1, \ldots, T_m) \), where \( T_k \) is the \( k \)-th component of \( T \). This map is an isomorphism. 5. For \( V_m \) defined as \( V \times \ldots \times V \) (\( m \) times), we can define an isomorphism \( \Phi: V_m \rightarrow L(\mathbb{F}^m, V) \) by \( \Phi((v_1, \ldots, v_m))(\lambda_1, \ldots, \lambda_m) = \lambda_1 v_1 + \ldots + \lambda_m v_m \). 6. Given \( v + U = x + W \), let \( u = x - v \). Then \( u \in U \) and \( x = v + u \). If \( v + U = x + W \) implies \( U = W \), then \( u \in W \), and \( U = W \). 7. Let \( U = \{(x, y, z) \in \mathbb{R}^3 : 2x + 3y + 5z = 0\} \). To show that \( A \) is a translate of \( U \) if and only if \( A = \{(x, y, z) \in \mathbb{R}^3 : 2x + 3y + 5z = c\} \), where \( c \) is a constant, consider the translation \( A + (0, 0, c) \). 8. (a) Let \( T \in L(V, W) \) and \( c \in W \). The set \( \{x \in V : T(x) = c\} \) is either the empty set or a translate of \( \text{null} T \). (b) The set of solutions to a system of linear equations is either the empty set or a translate of some subspace of \( \mathbb{F}^n \). 9. A nonempty subset \( A \) of \( V \) is a translate of some subspace of \( V \) if and only if \( \lambda v + (1 - \lambda) w \in A \) for all \( v, w \in A \) and \( \lambda \in \mathbb{F} \). 10. Let \( A_1 = v + U_1 \) and \( A_2 = w + U_2 \). The intersection \( A_1 \cap A_2 \) is either a translate of some subspace of \( V \) or the empty set. 11. (a) Define \( U = \{(x_1, x_2, \ldots) \in \mathbb{F}^{\infty} : x_k \neq 0 \text{ for only finitely many } k\} \). Show that \( U \) is a subspace of \( \mathbb{F}^{\infty} \). (b) Prove that \( \mathbb{F}^{\infty}/U \) is infinite-dimensional. 12. (a) Prove that \( A = \{\lambda_1 v_1 + \ldots + \lambda_m v_m : \lambda_1, \ldots, \lambda_m \in \mathbb{F} \text{ and } \lambda_1 + \ldots + \lambda_m = 1\} \) is a translate of some subspace of \( V \). (b) If \( B \) is a translate of some subspace of \( V \) and \( \{v_1, \ldots, v_m\} \subset B \), then \( A \subset B \). (c) Prove that \( A \) is a translate of some subspace of \( V \) with dimension less than \( m \). 13. Given \( U \) such that \( V/U \) is finite-dimensional, define \( S: V/U \rightarrow V \) by \( S(v + U) = v \). Prove that \( S \) is an isomorphism. 14. Given \( W_i \) as a basis of \( V \), \( \{w_1 + U, \ldots, w_m + U\} \) forms a basis of \( V/U \). 15. Given \( \{v_1 + U, \ldots, v_m + U\} \) as a basis of \( V/U \) and \( \{u_1, \ldots, u_n\} \) as a basis of \( U \), \( \{v_1, \ldots, v_m, u_1, \ldots, u_n\} \) forms a basis of \( V \). 16. For \( \varphi \in L(V, \mathbb{F}) \), define \( U = \text{null} \varphi \). Prove that \( \text{dim} V/U = 1 \). 17. Given \( U \) such that \( \text{dim} V/U = 1 \), define \( \varphi: V \rightarrow \mathbb{F} \) by \( \varphi(v + U) = \lambda \), where \( \lambda \) is the unique scalar such that \( v + U \) is in the one-dimensional subspace. 18. (a) Show that if \( W \) is a finite-dimensional subspace of \( V \) and \( V = U + W \), then \( \text{dim} W \geq \text{dim} V/U \). (b) Prove that there exists a finite-dimensional subspace \( W \) of \( V \) such that \( \text{dim} W = \text{dim} V/U \) and \( V = U \oplus W \). 19. Given \( T \in L(V, W) \) and \( U \) is a subspace of \( V \), the quotient map \( \pi: V \rightarrow V/U \) can be used to define \( S: V/U \rightarrow W \) by \( S(v + U) = T(v) \). Prove that \( T = S \circ \pi \) if and only if \( U \subset \text{null} T \). 20. \end{array} $