Reference no: EM133338346

Question 1. (NS 2.5.5) Consider the standard basis E , and the basis B = {(1, 2, 3), (3, 2, 1), (0, 0, 1)} for R³.

(a) Compute the change of basis matrices [I]_B,E and [I]_E,B.

(b) Determine the matrices [T]_B,B, [T]_B,E , [T]_E,B , and [T]_E,E for the linear operator T on R³ given by T (a, b, c) = (6a + b, a - b - c, 2a - b + 2c).

Question 2. Let V be the vector space of all differentiable functions from R to R. Let B = {e^5t, e^5tcos t, e^5t sin t} and consider the subspace W = span(B) of V.

(a) Let D be the differentation operator (i.e., D(f (t)) = f'(t)) on W. Find the matrix representation [D]_B,B.
(b) Verify Theorem 2.41 (i.e., [Df]_B = [D]_B,B [f ]_B) for the input function f (t) = 3e^5t - e^5tcos t + 2e^5tsin t.

Question 3. (variation on NS 2.5.6) Let B := {u₁, . . . , u_n} and C := {v₁, . . . , v_n} be ordered bases for a vector space V . Let T be a linear operator on V defined by Tu₁ = v₁, . . . , Tu_nv_n. Show that [T]_B,B = [I ]_B,C.
(Note. There is a typo in the statement of this problem in the book.)

Question 4. Let A and B be square matrices. Prove or disprove each of the following:
(a) If A is equivalent to B, then A² is equivalent to B².
(b) If A is similar to B, then A² is similar to B².