(Rank-1 Updates of Linear Systems) (a) Set A = round(10 rand(8)) b = round(10 rand(8

(Change of Basis) Set U = round(20 rand(4)) 10, V = round(10 rand(4)) and set b = ones(4, 1). (a) We can use the MATLAB function rank to determine whether the column vectors of a matrix are linearly independent. What should the rank be if the column vectors of U are linearly independent? Compute the rank of U, and verify that its column vectors are linearly independent and hence form a basis for R4. Compute the rank of V, and verify that its column vectors also form a basis for R4. (b) Use MATLAB to compute the transition matrix from the standard basis for R4 to the ordered basis E = {u1, u2, u3, u4}. [Note that in MATLAB the notation for the jth column vector uj is U(:, j).] Use this transition matrix to compute the coordinate vector c of b with respect to E. Verify that b = c1u1 + c2u2 + c3u3 + c4u4 = Uc (c) Use MATLAB to compute the transition matrix from the standard basis to the ordered basis F = {v1, v2, v3, v4}, and use this transition matrix to find the coordinate vector d of b with respect to F. Verify that b = d1v1 + d2v2 + d3v3 + d4v4 = Vd (d) Use MATLAB to compute the transition matrix S from E to F and the transition matrix T from F to E. How are S and T related? Verify that Sc = d and Td = c.

(Rank-Deficient Matrices) In this exercise we consider how to use MATLAB to generate matrices with specified ranks. (a) In general, if A is an m n matrix with rank r, then r min(m, n). Why? Explain. If the entries of A are random numbers, we would expect that r = min(m, n). Why? Explain. Check this out by generating random 6 6, 8 6, and 5 8 matrices and using the MATLAB command rank to compute their ranks. Whenever the rank of an mn matrix equals min(m, n), we say that the matrix has full rank. Otherwise, we say that the matrix is rank deficient. (b) MATLABs rand and round commands can be used to generate random m n matrices with integer entries in a given range [a, b]. This can be done with a command of the form A = round((b a) rand(m, n)) + a For example, the command A = round(4 rand(6, 8)) + 3 will generate a 6 8 matrix whose entries are random integers in the range from 3 to 7. Using the range [1, 10], create random integer 10 7, 8 12, and 10 15 matrices and in each case check the rank of the matrix. Do these integer matrices all have full rank? (c) Suppose that we want to use MATLAB to generate matrices with less than full rank. It is easy to generate matrices of rank 1. If x and y are nonzero vectors in Rm and Rn, respectively, then A = xyT will be an m n matrix with rank 1. Why? Explain. Verify this in MATLAB by setting x = round(9 rand(8, 1)) + 1, y = round(9 rand(6, 1)) + 1 and using these vectors to construct an 86 matrix A. Check the rank of A with the MATLAB command rank. (d) In general, rank(AB) min(rank(A), rank(B)) (1) (See Exercise 28 in Section 3.6.) If A and B are noninteger random matrices, the relation (1) should be an equality. Generate an 8 6 matrix A by setting X = rand(8, 2), Y = rand(6, 2), A = X Y What would you expect the rank of A to be? Explain. Test the rank of A with MATLAB. (e) Use MATLAB to generate matrices A, B, and C such that (i) A is 8 8 with rank 3. (ii) B is 6 9 with rank 4. (iii) C is 10 7 with rank 5.

. (Column Space and Reduced Row Echelon Form) Set B = round(10 rand(8, 4)) X = round(10 rand(4, 3)) C = B X A = [ B C ] (a) How are the column spaces of B and C related? (See Exercise 28 in Section 3.6.) What would you expect the rank of A to be? Explain. Use MATLAB to check your answer. (b) Which column vectors of A should form a basis for its column space? Explain. If U is the reduced row echelon form of A, what would you expect its first four columns to be? Explain. What would you expect its last four rows to be? Explain. Use MATLAB to verify your answers by computing U. (c) Use MATLAB to construct another matrix D = ( E EY ), where E is a random 6 4 matrix and Y is a random 4 2 matrix. What would you expect the reduced row echelon form of D to be? Compute it with MATLAB. Show that, in general, if B is an m n matrix of rank n and X is an nk matrix, the reduced row echelon form of ( B BX ) will have block structure (I X ) if m = n or I X O O if m > n

(Rank-1 Updates of Linear Systems) (a) Set A = round(10 rand(8)) b = round(10 rand(8, 1)) M = inv(A) Use the matrix M to solve the system Ay = b for y. (b) Consider now a new system Cx = b, where C is constructed as follows: u = round(10 rand(8, 1)) v = round(10 rand(8, 1)) E = u v C = A + E The matrices C and A differ by the rank-1 matrix E. Use MATLAB to verify that the rank of E is 1. Use MATLABs \ operation to solve the system Cx = b and then compute the residual vector r = b Ax. (c) Let us now solve Cx = b by a new method that takes advantage of the fact that A and C differ by a rank-1 matrix. This new procedure is called a rank-1 update method. Set z = M u, c = v  y, d = v  z, e = c/(1 + d) and then compute the solution x by x = y e z Compute the residual vector b Cx and compare it with the residual vector in part (b). This new method may seem more complicated, but it actually is much more computationally efficient. (d) To see why the rank-1 update method works, use MATLAB to compute and compare Cy and b + cu Prove that if all computations had been carried out in exact arithmetic, these two vectors would be equal. Also, compute Cz and (1 + d)u Prove that if all computations had been carried out in exact arithmetic, these two vectors would be equal. Use these identities to prove that Cx = b. Assuming that A is nonsingular, will the rank-1 update method always work? Under what conditions could it fail? Explain.

True or FalseIf S and T are subspaces of a vector space V, then S T is a subspace of V.

True or False If x1, x2, ... , xn span Rn, then they are linearly independent.

True or FalseIf x1, x2, ... , xn span a vector space V, then they are linearly independent.

True or FalseIf x1, x2, ... , xk are vectors in a vector space V and Span(x1, x2, ... , xk) = Span(x1, x2, ... , xk1) then x1, x2, ... , xk are linearly dependent.

True or False If A is an m n matrix, then A and AT have the same rank.

True or FalseIf A is an m n matrix, then A and AT have the same nullity.

True or False . If U is the reduced row echelon form of A, then A and U have the same row space.

True or FalseIf U is the reduced row echelon form of A, then A and U have the same column space.

True or FalseLet x1, x2,. . . , xk be linearly independent vectors in Rn. If k < n and xk+1 is a vector that is not in Span(x1, x2, ... , xk), then the vectors x1, x2,. . . , xk, xk+1 are linearly independent.

True or False Let {u1, u2}, {v1, v2}, and {w1, w2}, be bases for R2. If X is the transition matrix corresponding to a change of basis from {u1, u2} to {v1, v2} and Y is the transition matrix corresponding to a change of basis from {v1, v2} to {w1, w2}, then Z = XY is the transition matrix corresponding to the change of basis from {u1, u2} to {w1, w2}.

True or False If A and B are n n matrices that have the same rank, then the rank of A2 must equal the rank of B2.