Let B = {B]. A} mation x→→ Ax. =[%] be a basis for R2 and A = Find the B-matrix for the transfor-

9. Consider the following 2 x 2 matrix A and basis S of R²: A ¹ = [² %) and 5 = {[_¹₂] · [_³;]} The matrix A defines a linear operator on R². Find a matrix B that represents the mapping A relative to the basis S.

Label the following statements as true or false. (a) Suppose that B = {x1,x2, ..., xn} and B' = {x1, x2, ... , x,} are ordered bases for a vector space and Q is the change of coordinate matrix that changes B'-coordinates into 3-coordinates. Then the jth column of Q is [x;]g. (b) Every change of coordinate matrix is invertible. (c) Let T be a linear operator on a finite-dimensional vector space V, let 3 and B' be ordered bases for V, and let Q be the change of coordinate matrix that changes B'-coordinates into 3-coordinates. Then [T] = Q[T]8•Q¬!.

Consider the linear transformation T: R^ →R" whose matrix A relative to the standard basis is given. [³ A = (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (1₁, 12) = 4,5 B₂ 3-2 (b) Find a basis for each of the corresponding eigenspaces. B₁ = 1,2 128 1,1 = A' = 1 6 (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). X 5 5 1 2 ↓ 1

Consider the vector space V of all the real polynomials of degree less than or equal to three,  p(x) = ax^3 + b x^2 + c x + d, and consider the linear transformation T: V ------> V given by the derivative, i.e., T( p ) = p' = 3ax^2 + 2 b x + c. Find the eigenvalues and eigenvectors of this transformation.