10. Let P denote the vector space of polynomials of degree at most k. Let T: P3 → P2 be the function defined by T (p(x)) = p"(x) - 2p' (r). This is & linear map. (a) Find the matrix for T with respect to the standard bases for P3 and P2. (b) Find a basis for ker T.

Let T: R² →→ R² where Find the matrix A such that T A = ([₁]) = ^ [*] · A T([7])=[52] and 1([2])-[1] -59 -43 T 469 238

4. Let W be the xy-plane in R³. (a) Find an orthonormal basis ū₁, 2 for W. (b) Construct the matrix Q = [ ₁ ü₂ ] and calculate the matrix QQT. (c) Consider the linear transformation T: R³ → R³ that projects each vector orthogonally onto the xy-plane. Note that T() = QQT. Use the matrix to calculate T(₁), T(2), and T(ē3). (d) Explain how your answers make sense geometrically.

Find the matrix A' for T relative to the basis B'. T: R² A' = X A' = 3 11/3 R², T(x, y) = (x - y, y - 2x), B' = {(1, -2), (0, 3)} -3 48 -16 -4 Find the matrix A' for T relative to the basis B'. ↓ 1 T: R² → R², T(x, y) = (-7x + y, 7x - y), B' = {(1, −1), (-1,5)} -72 24