# Set A = triu(ones(5)) tril(ones(5)). If L denotes the linear operator defined by L(x) =

Chapter 4, Problem 2

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Set A = triu(ones(5)) tril(ones(5)). If L denotes the linear operator defined by L(x) = Ax for all x in Rn, then A is the matrix representing L with respect to the standard basis for R5. Construct a 5 5 matrix U by setting U = hankel(ones(5, 1), 1 : 5) Use the MATLAB function rank to verify that the column vectors of U are linearly independent. Thus, E = {u1, u2, u3, u4, u5} is an ordered basis for R5. The matrix U is the transition matrix from E to the standard basis. (a) Use MATLAB to compute the matrix B representing L with respect to E. (The matrix B should be computed in terms of A, U, and U1.) (b) Generate another matrix by setting V = toeplitz([1, 0, 1, 1, 1]) Use MATLAB to check that V is nonsingular. It follows that the column vectors of V are linearly independent and hence form an ordered basis F for R5. Use MATLAB to compute the matrix C, which represents L with respect to F. (The matrix C should be computed in terms of A, V, and V1.) (c) The matrices B and C from parts (a) and (b) should be similar. Why? Explain. Use MATLAB to compute the transition matrix S from F to E. Compute the matrix C in terms of B, S, and S1. Compare your result with the result from part (b).

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