Let A and B be n n matrices. (a) Show that AB = O if and only if the column space of B

For each of the following matrices, find a basis for the row space, a basis for the column space, and a basis for the null space: (a) 1 3 2 2 1 4 4 7 8 (b) 3 1 3 4 1 2 1 2 3 8 4 2 (c) 1 3 2 1 2 1 3 2 3 4 5 6

In each of the following, determine the dimension of the subspace of R3 spanned by the given vectors: (a) 1 2 2 , 2 2 4 , 3 3 6 (b) 1 1 1 , 1 2 3 , 2 3 1 (c) 1 1 2 , 2 2 4 , 3 2 5 , 2 1 3

Let A = 1 2 2 3 1 4 2 4 5 5 4 9 3 6 7 8 5 9 (a) Compute the reduced row echelon form U of A. Which column vectors of U correspond to the free variables? Write each of these vectors as a linear combination of the column vectors corresponding to the lead variables. (b) Which column vectors of A correspond to the lead variables of U? These column vectors form a basis for the column space of A. Write each of the remaining column vectors of A as a linear combination of these basis vectors.

For each of the following choices of A and b, determine whether b is in the column space of A and state whether the system Ax = b is consistent: (a) A = 1 2 2 4 , b = 4 8 (b) A = 3 6 1 2 , b = 1 1 (c) A = 2 1 3 4 , b = 4 6 (d) A = 1 1 2 1 1 2 1 1 2 , b = 1 2 3 (e) A = 0 1 1 0 0 1 , b = 2 5 2 (f) A = 1 2 2 4 1 2 , b = 5 10 5

For each consistent system in Exercise 4, determine whether there will be one or infinitely many solutions by examining the column vectors of the coefficient matrix A.

How many solutions will the linear system Ax = b have if b is in the column space of A and the column vectors of A are linearly dependent? Explain.

Let A be an 6 n matrix of rank r and let b be a vector in R6. For each pair of values of r and n that follow, indicate the possibilities as to the number of solutions one could have for the linear system Ax = b. Explain your answers. (a) n = 7, r = 5 (b) n = 7, r = 6 (c) n = 5, r = 5 (d) n = 5, r = 4

Let A be an m n matrix with m > n. Let b Rm and suppose that N(A) = {0}. (a) What can you conclude about the column vectors of A? Are they linearly independent? Do they span Rm? Explain. (b) How many solutions will the system Ax = b have if b is not in the column space of A? How many solutions will there be if b is in the column space of A? Explain.

Let A and B be 6 5 matrices. If dim N(A) = 2, what is the rank of A? If the rank of B is 4, what is the dimension of N(B)? 1

Let A be an m n matrix whose rank is equal to n. If Ac = Ad, does this imply that c must be equal to d? What if the rank of A is less than n? Explain your answers. 1

Let A be an m n matrix. Prove that rank(A) min(m, n) 1

Let A and B be row-equivalent matrices. (a) Show that the dimension of the column space of A equals the dimension of the column space of B. (b) Are the column spaces of the two matrices necessarily the same? Justify your answer. 1

Let A be a 4 3 matrix and suppose that the vectors z1 = 1 1 2 , z2 = 1 0 1 form a basis for N(A). If b = a1 + 2a2 + a3, find all solutions of the system Ax = b. 1

Let A be a 4 4 matrix with reduced row echelon form given by U = 1 0 2 1 0 1 1 4 0 0 0 0 0 0 0 0 If a1 = 3 5 2 1 and a2 = 4 3 7 1 find a3 and a4. 1

Let A be a 4 5 matrix and let U be the reduced row echelon form of A. If a1 = 2 1 3 2 , a2 = 1 2 3 1 , U = 1 0 2 0 1 0 1 3 0 2 0 0 0 1 5 0 0 0 0 0 (a) find a basis for N(A). (b) given that x0 is a solution of Ax = b, where b = 0 5 3 4 and x0 = 3 2 0 2 0 (i) find all solutions to the system. (ii) determine the remaining column vectors of A. 1

Let A be a 5 8 matrix with rank equal to 5 and let b be any vector in R5. Explain why the system Ax = b must have infinitely many solutions. 1

Let A be a 4 5 matrix. If a1, a2, and a4 are linearly independent and a3 = a1 + 2a2, a5 = 2a1 a2 + 3a4 determine the reduced row echelon form of A. 1

Let A be a 5 3 matrix of rank 3 and let {x1, x2, x3} be a basis for R3. (a) Show that N(A) = {0}. (b) Show that if y1 = Ax1, y2 = Ax2, y3 = Ax3, then y1, y2, and y3 are linearly independent. (c) Do the vectors y1, y2, y3 from part (b) form a basis for R5? Explain. 1

Let A be an m n matrix with rank equal to n. Show that if x _= 0 and y = Ax, then y _= 0. 2

Prove that a linear system Ax = b is consistent if and only if the rank of (A | b) equals the rank of A. 2

Let A and B be m n matrices. Show that rank(A + B) rank(A) + rank(B) 2

Let A be an m n matrix. (a) Show that if B is a nonsingular m m matrix, then BA and A have the same null space and hence the same rank. (b) Show that if C is a nonsingular n n matrix, then AC and A have the same rank. 2

Prove Corollary 3.6.4. 2

Show that if A and B are n n matrices and N(A B) = Rn, then A = B. 2

Let A and B be n n matrices. (a) Show that AB = O if and only if the column space of B is a subspace of the null space of A. (b) Show that if AB = O, then the sum of the ranks of A and B cannot exceed n. 2

Let A Rmn and b Rm, and let x0 be a particular solution of the system Ax = b. Prove the following: (a) A vector y in Rn will be a solution of Ax = b if and only if y = x0 + z, where z N(A). (b) If N(A) = {0}, then the solution x0 is unique. 2

Let x and y be nonzero vectors in Rm and Rn, respectively, and let A = xyT . (a) Show that {x} is a basis for the column space of A and that {yT } is a basis for the row space of A. (b) What is the dimension of N(A)? 2

Let A Rmn, B Rnr , and C = AB. Show that (a) the column space of C is a subspace of the column space of A. (b) the row space of C is a subspace of the row space of B. (c) rank(C) min{rank(A), rank(B)}. 2

Let A Rmn, B Rnr , and C = AB. Show that (a) if A and B both have linearly independent column vectors, then the column vectors of C will also be linearly independent. (b) if A and B both have linearly independent row vectors, then the row vectors of C will also be linearly independent. [Hint: Apply part (a) to CT .] 3

Let A Rmn, B Rnr , and C = AB. Show that (a) if the column vectors of B are linearly dependent, then the column vectors of C must be linearly dependent. (b) if the row vectors of A are linearly dependent, then the row vectors of C are linearly dependent. [Hint: Apply part (a) to CT .] 3

An m n matrix A is said to have a right inverse if there exists an n m matrix C such that AC = Im. A is said to have a left inverse if there exists an n m matrix D such that DA = In. (a) Show that if A has a right inverse, then the column vectors of A span Rm. (b) Is it possible for an mn matrix to have a right inverse if n < m? n m? Explain. 3

Prove: If A is an m n matrix and the column vectors of A span Rm, then A has a right inverse. [Hint: Let ej denote the j th column of Im, and solve Ax = ej for j = 1, . . . ,m.] 3

Show that a matrix B has a left inverse if and only if BT has a right inverse. 3

Let B be an n m matrix whose columns are linearly independent. Show that B has a left inverse. 3

Prove that if a matrix B has a left inverse, then the columns of B are linearly independent. 3

Show that if a matrix U is in row echelon form, then the nonzero row vectors of U form a basis for the row space of U.