$ \begin{array}{l} \text{Exercises 3D} \\ \\ 1 \text{ Suppose } T \in L(V, W) \text{ is invertible. Show that } T^{-1} \text{ is invertible and } (T^{-1})^{-1} = T. \\ \\ 2 \text{ Suppose } T \in L(U, V) \text{ and } S \in L(V, W) \text{ are both invertible linear maps. Prove that } ST \in L(U, W) \text{ is invertible and that } (ST)^{-1} = T^{-1} S^{-1}. \\ \\ 3 \text{ Suppose } V \text{ is finite-dimensional and } T \in L(V). \text{ Prove that the following are equivalent.} \\ \text{(a) } T \text{ is invertible.} \\ \text{(b) } T v_1, ..., T v_n \text{ is a basis of } V \text{ for every basis } v_1, ..., v_n \text{ of } V. \\ \text{(c) } T v_1, ..., T v_n \text{ is a basis of } V \text{ for some basis } v_1, ..., v_n \text{ of } V. \\ \\ 4 \text{ Suppose } V \text{ is finite-dimensional and } \text{dim } V > 1. \text{ Prove that the set of noninvertible linear maps from } V \text{ to itself is not a subspace of } L(V). \\ \\ 5 \text{ Suppose } V \text{ is finite-dimensional, } U \text{ is a subspace of } V, \text{ and } S \in L(U, V). \text{ Prove that there exists an invertible linear map } T \text{ from } V \text{ to itself} \\ \text{ such that } Tu = Su \text{ for every } u \in U \text{ if and only if } S \text{ is injective.} \\ \\ 6 \text{ Suppose that } W \text{ is finite-dimensional and } S, T \in L(V, W). \text{ Prove that } \text{null } S = \text{null } T \text{ if and only if there exists an invertible } E \in L(W) \text{ such that } S = ET. \\ \\ 7 \text{ Suppose that } V \text{ is finite-dimensional and } S, T \in L(V, W). \text{ Prove that } \text{range } S = \text{range } T \text{ if and only if there exists an invertible } E \in L(V) \text{ such that } S = TE. \\ \\ 8 \text{ Suppose } V \text{ and } W \text{ are finite-dimensional and } S, T \in L(V, W). \text{ Prove that there exist invertible } E_1 \in L(V) \text{ and } E_2 \in L(W) \\ \text{ such that } S = E_2 T E_1 \text{ if and only if } \text{dim null } S = \text{dim null } T. \\ \\ 9 \text{ Suppose } V \text{ is finite-dimensional and } T \colon V \rightarrow W \text{ is a surjective linear map of } V \text{ onto } W. \text{ Prove that there is a subspace } U \text{ of } V \\ \text{ such that } T|_U \text{ is an isomorphism of } U \text{ onto } W. \\ \text{ Here } T|_U \text{ means the function } T \text{ restricted to } U. \text{ Thus } T|_U \text{ is the function whose domain is } U, \text{ with } T|_U \text{ defined by } T|_U(u) = Tu \text{ for every } u \in U. \\ \\ 10 \text{ Suppose } V \text{ and } W \text{ are finite-dimensional and } U \text{ is a subspace of } V. \text{ Let } E = \{ T \in L(V, W) \colon U \subseteq \text{ null } T \}. \\ \text{(a) Show that } E \text{ is a subspace of } L(V, W). \\ \text{(b) Find a formula for } \text{dim } E \text{ in terms of } \text{dim } V, \text{ dim } W, \text{ and } \text{dim } U. \\ \text{Hint: Define } \Phi \colon L(V, W) \rightarrow L(U, W) \text{ by } \Phi(T) = T|_U. \text{ What is } \text{null } \Phi? \text{ What is } \text{range } \Phi? \\ \\ 11 \text{ Suppose } V \text{ is finite-dimensional and } S, T \in L(V). \text{ Prove that } ST \text{ is invertible } \iff S \text{ and } T \text{ are invertible.} \\ \\ 12 \text{ Suppose } V \text{ is finite-dimensional and } S, T, U \in L(V) \text{ and } STU = I. \text{ Show that } T \text{ is invertible and that } T^{-1} = US. \\ \\ 13 \text{ Show that the result in Exercise 12 can fail without the hypothesis that } V \text{ is finite-dimensional.} \\ \\ 14 \text{ Prove or give a counterexample: If } V \text{ is a finite-dimensional vector space and } R, S, T \in L(V) \text{ are such that } RTS \text{ is surjective, then } S \text{ is injective.} \\ \\ 15 \text{ Suppose } T \in L(V) \text{ and } v_1, ..., v_m \text{ is a list in } V \text{ such that } Tv_1, ..., Tv_m \text{ spans } V. \text{ Prove that } v_1, ..., v_m \text{ spans } V. \\ \\ 16 \text{ Prove that every linear map from } \mathbb{F}^{n,1} \text{ to } \mathbb{F}^{m,1} \text{ is given by a matrix multiplication. In other words, prove that if } T \in L(\mathbb{F}^{n,1}, \mathbb{F}^{m,1}), \text{ then there exists an } m \times n \text{ matrix } A \text{ such that } Tx = Ax \text{ for every } x \in \mathbb{F}^{n,1}. \\ \\ \text{Section 3D Invertibility and Isomorphisms} \\ \\ 17 \text{ Suppose } V \text{ is finite-dimensional and } S \in L(V). \text{ Define } \mathcal{A} \in L(L(V)) \text{ by} \\ \mathcal{A}(T) = ST \\ \text{(a) Show that } \text{dim null } \mathcal{A} = (\text{dim } V)(\text{dim null } S). \\ \text{(b) Show that } \text{dim range } \mathcal{A} = (\text{dim } V)(\text{dim range } S). \\ \\ 18 \text{ Show that } V \text{ and } L(\mathbb{F}, V) \text{ are isomorphic vector spaces.} \\ \\ 19 \text{ Suppose } V \text{ is finite-dimensional and } T \in L(V). \text{ Prove that } T \text{ has the same matrix with respect to every basis of } V \text{ if and only if } T \text{ is a scalar multiple of the identity operator.} \\ \\ 20 \text{ Suppose } q \in \mathbb{P}(\mathbb{R}). \text{ Prove that there exists a polynomial } p \in \mathbb{P}(\mathbb{R}) \text{ such that} \\ q(x) = (x^2 + x) p''(x) + 2x p'(x) + p'(3) \\ \text{ for all } x \in \mathbb{R}. \\ \\ 21 \text{ Suppose } n \text{ is a positive integer and } A_{j,k} \in \mathbb{F} \text{ for all } j, k = 1, ..., n. \text{ Prove that the following are equivalent (note that in both parts below, the number of equations equals the number of variables).} \\ \text{(a) The trivial solution } x_1 = \cdots = x_n = 0 \text{ is the only solution to the} \\ \text{homogeneous system of equations} \\ \sum_{k=1}^n A_{1,k} x_k = 0 \\ \vdots \\ \sum_{k=1}^n A_{n,k} x_k = 0. \\ \text{(b) For every } c_1, ..., c_n \in \mathbb{F}, \text{ there exists a solution to the system of equations} \\ \sum_{k=1}^n A_{1,k} x_k = c_1 \\ \vdots \\ \sum_{k=1}^n A_{n,k} x_k = c_n. \\ \\ 22 \text{ Suppose } T \in L(V) \text{ and } v_1, ..., v_n \text{ is a basis of } V. \text{ Prove that} \\ M(T,(v_1,...,v_n)) \text{ is invertible } \iff T \text{ is invertible.} \\ \\ 23 \text{ Suppose that } u_1, ..., u_n \text{ and } v_1, ..., v_n \text{ are bases of } V. \text{ Let } T \in L(V) \text{ be such} \\ \text{ that } Tv_k = u_k \text{ for each } k = 1,...,n. \text{ Prove that} \\ M(T, (v_1, ..., v_n)) = M(I, (u_1, ..., u_n), (v_1, ..., v_n)). \\ \\ 24 \text{ Suppose } A \text{ and } B \text{ are square matrices of the same size and } AB = I. \text{ Prove that } BA = I. \end{array} $