Let E = {x1, x2, . . . , xn} be an ordered basis for Rn. If L1 : Rn Rn and L2 : Rn Rn

Use MATLAB to generate a matrix W and a vector x by setting W = triu(ones(5)) and x = [1 : 5]_ The columns of W can be used to form an ordered basis F = {w1,w2,w3,w4,w5} Let L : R5 R5 be a linear operator such that L(w1) = w2, L(w2) = w3, L(w3) = w4 and L(w4) = 4w1 + 3w2 + 2w3 + w4 L(w5) = w1 + w2 + w3 + 3w4 + w5 (a) Determine the matrix A representing L with respect to F, and enter it in MATLAB. (b) Use MATLAB to compute the coordinate vector y = W1x of x with respect to F. (c) Use A to compute the coordinate vector z of L(x) with respect to F. (d) W is the transition matrix from F to the standard basis for R5. Use W to compute the coordinate vector of L(x) with respect to the standard basis.

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).

Let A = toeplitz(1 : 7), S = compan(ones(8, 1)) and set B = S1 A S. The matrices A and B are similar. Use MATLAB to verify that the following properties hold for these two matrices: (a) det(B) = det(A) (b) BT = ST AT (ST ) 1 (c) B1 = S1A1S (d) B9 = S1A9S (e) B 3I = S1(A 3I )S (f) det(B 3I ) = det(A 3I ) (g) tr(B) = tr(A) (Note that the trace of a matrix A can be computed with the MATLAB command trace.) These properties will hold in general for any pair of similar matrices (see Exercises 1115 of Section 3).

If L1 rotates each vector x in R2 by 60 and then reflects the resulting vector about the x-axis, and if L2 is a transformation that does the same two operations, but in the reverse order, then L1 = L2.

The set of all vectors x used in the homogeneous coordinate system (see the application on computer graphics and animation in Section 2 ) forms a subspace of R3.

Let L : R2 R2 be a linear operator, and let A be the standard matrix representation of L. If L2 is defined by L2(x) = L(L(x)) for all x R2 then L2 is a linear operator and its standard matrix representation is A2.

Let E = {x1, x2, . . . , xn} be an ordered basis for Rn. If L1 : Rn Rn and L2 : Rn Rn have the same matrix representation with respect to E, then L1 = L2.

Let L : Rn Rn be a linear operator. If A is the standard matrix representation of L, then an n n matrix B will also be a matrix representation of L if and only if B is similar to A.

Let A, B, and C be n n matrices. If A is similar to B and B is similar to C, then A is similar to C. 1

Any two matrices with the same trace are similar. [This statement is the converse of part (b) of Exercise 15 in Section 3.]