Chapter 4, Problem 2E

Set $A=triu(ones(5))*tril(ones(5))$ . If L denotes the linear operator defined by $L(x)=Ax$ for all x in ${ℝ}^n$ , then A is the matrix representing L with respect to the standard basis for ${ℝ}^5$ . Construct $5\times 5$ 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, $E=\{u_1,u_2,u_3,u_4,u_5\}$ is an ordered basis for ${ℝ}^5$ . 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 $U^{-1}$ .)
(b) Generate another matrix by setting
$V=toeplitz([1,0,1,1,1])$
Use MATL.AB 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 ${ℝ}^5$ . 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 $V^{-1}$ .)
(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 $S^{-1}$ . Compare your result with the result from part (b).