$ \begin{array}{l} \text{Exercises 3C} \\ \\ 1 \text{ Suppose } T \in L(V, W). \text{ Show that with respect to each choice of bases of } V \text{ and } W, \text{ the matrix of } T \text{ has at least dim range } T \text{ nonzero entries.} \\ \\ 2 \text{ Suppose } V \text{ and } W \text{ are finite-dimensional and } T \in L(V,W). \text{ Prove that } \text{dim range } T = 1 \text{ if and only if there exist a basis of } V \text{ and a basis of } W \\ \text{ such that with respect to these bases, all entries of } M(T) \text{ equal } 1. \\ \\ 3 \text{ Suppose } v_1,...,v_n \text{ is a basis of } V \text{ and } w_1,...,w_m \text{ is a basis of } W. \\ \text{(a) Show that if } S, T \in L(V,W), \text{ then } M(S+T) = M(S) + M(T). \\ \text{(b) Show that if } \lambda \in \mathbb{F} \text{ and } T \in L(V,W), \text{ then } M(\lambda T) = \lambda M(T). \\ \\ 4 \text{ Suppose that } D \in L(\mathbb{P}_3(\mathbb{R}), \mathbb{P}_2(\mathbb{R})) \text{ is the differentiation map defined by } Dp = p'. \text{ Find a basis of } \mathbb{P}_3(\mathbb{R}) \text{ and a basis of } \mathbb{P}_2(\mathbb{R}) \\ \text{ such that the matrix of } D \text{ with respect to these bases is} \\ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{pmatrix}. \\ \text{Compare with Example 3.33. The next exercise generalizes this exercise.} \\ \\ 5 \text{ Suppose } V \text{ and } W \text{ are finite-dimensional and } T \in L(V, W). \text{ Prove that there exist a basis of } V \text{ and a basis of } W \\ \text{ such that with respect to these bases, all entries of } M(T) \text{ are } 0 \text{ except that the entries in row } k, \text{ column } k, \text{ equal } 1 \\ \text{ if } 1 \leq k \leq \text{dim range } T. \\ \\ 6 \text{ Suppose } v_1, ..., v_m \text{ is a basis of } V \text{ and } W \text{ is finite-dimensional. Suppose } T \in L(V, W). \text{ Prove that there exists a basis } \\ w_1, ..., w_n \text{ of } W \text{ such that all entries in the first column of } M(T) \text{ [with respect to the bases } v_1, ..., v_m \text{ and } w_1, ..., w_n] \text{ are } 0 \\ \text{ except for possibly a } 1 \text{ in the first row, first column.} \\ \text{In this exercise, unlike Exercise 5, you are given the basis of } V \text{ instead of being able to choose a basis of } V. \\ \\ 7 \text{ Suppose } w_1, ..., w_n \text{ is a basis of } W \text{ and } V \text{ is finite-dimensional. Suppose } T \in L(V, W). \text{ Prove that there exists a basis } \\ v_1, ..., v_m \text{ of } V \text{ such that all entries in the first row of } M(T) \text{ [with respect to the bases } v_1, ..., v_m \text{ and } w_1, ..., w_n] \text{ are } 0 \\ \text{ except for possibly a } 1 \text{ in the first row, first column.} \\ \text{In this exercise, unlike Exercise 5, you are given the basis of } W \text{ instead of being able to choose a basis of } W. \\ \\ 8 \text{ Suppose } A \text{ is an } m \times n \text{ matrix and } B \text{ is an } n \times p \text{ matrix. Prove that } (AB)_{j,\cdot} = A_{j,\cdot} B \text{ for each } 1 \leq j \leq m. \text{ In other words, show that row } j \text{ of } AB \\ \text{ equals (row } j \text{ of } A) \text{ times } B. \\ \text{This exercise gives the row version of 3.48.} \\ \\ 9 \text{ Suppose } a = (a_1,...,a_n) \text{ is a } 1 \times n \text{ matrix and } B \text{ is an } n \times p \text{ matrix.} \\ \text{Prove that} \\ aB \text{ is a linear combination of the rows of } B, \text{ with } aB = a_1B_{1,\cdot} + \cdots + a_nB_{n,\cdot}. \text{ The scalars that multiply the rows coming from } a. \\ \text{This exercise gives the row version of 3.50.} \\ \\ 10 \text{ Give an example of } 2 \times 2 \text{ matrices } A \text{ and } B \text{ such that } AB \neq BA. \\ \\ 11 \text{ Prove that the distributive property holds for matrix addition and matrix multiplication. In other words, suppose } A, B, C, D, E, \text{ and } F \\ \text{ are matrices whose sizes are such that } A(B + C) \text{ and } (D + E)F \text{ make sense. Explain why } A B + A C \text{ and } D F + E F \text{ both make sense and prove that} \\ A(B + C) = A B + A C \text{ and } (D + E) F = D F + E F. \\ \\ 12 \text{ Prove that matrix multiplication is associative. In other words, suppose } A, B, \text{ and } C \text{ are matrices whose sizes are such that } (AB)C \text{ makes sense. Explain why } A (B C) \text{ makes sense and prove that} \\ (AB)C = A(B C). \\ \text{Try to find a clean proof that illustrates the following quote from Emil Artin: “It is my experience that proofs involving matrices can be shortened by 50%} \\ \text{if one throws the matrices out.”} \\ \\ 13 \text{ Suppose } A \text{ is an } n \times n \text{ matrix and } 1 \leq j, k \leq n. \text{ Show that the entry in row } j, \text{ column } k, \text{ of } A^3 \text{ (which is defined to mean } A A A) \text{ is} \\ \sum_{p=1}^n \sum_{r=1}^n A_{j,p} A_{p,r} A_{r,k}. \\ \\ 14 \text{ Suppose } m \text{ and } n \text{ are positive integers. Prove that the function } A \mapsto A^t \text{ is a linear map from } \mathbb{F}^{m,n} \text{ to } \mathbb{F}^{n,m}. \\ \\ \text{Section 3C Matrices} \\ \\ 15 \text{ Prove that if } A \text{ is an } m \times n \text{ matrix and } C \text{ is an } n \times p \text{ matrix, then} \\ (AB)^t = B^t A^t. \\ \text{This exercise shows that the transpose of the product of two matrices is the product of the transposes in the opposite order.} \\ \\ 16 \text{ Suppose } A \text{ is an } m \times n \text{ matrix with } A \neq 0. \text{ Prove that the rank of } A \text{ is } 1 \text{ if and only if there exist } (c_1, ..., c_m) \in \mathbb{F}^m \\ \text{ and } (d_1, ..., d_n) \in \mathbb{F}^n \text{ such that } A_{j,k} = c_j d_k \text{ for every } j = 1,...,m \text{ and every } k = 1,...,n. \\ \\ 17 \text{ Suppose } T \in L(V), \text{ and } u_1, ..., u_n \text{ and } v_1, ..., v_n \text{ are bases of } V. \text{ Prove that the following are equivalent.} \\ \text{(a) } T \text{ is injective.} \\ \text{(b) The columns of } M(T) \text{ are linearly independent in } \mathbb{F}^{n,1}. \\ \text{(c) The columns of } M(T) \text{ span } \mathbb{F}^{n,1}. \\ \text{(d) The rows of } M(T) \text{ span } \mathbb{F}^{1,n}. \\ \text{(e) The rows of } M(T) \text{ are linearly independent in } \mathbb{F}^{1,n}. \\ \text{Here } M(T) \text{ means } M(T, (u_1, ..., u_n), (v_1, ..., v_n)). \end{array} $