Let T: R² R² be the linear map T(x, y) = (y,x). (a) What is the standard matrix of T? (b) Give a geometric description of T using the word reflection. (c) Find the eigenvalues of T (just using the definition).

Let T : P2 → M2x2 be the linear transformation defined by T(ax? + bx + c) |0 a (a) Find the matrix for T with respect to the standard bases of P2 and M2x2- (b) Find the matrix for T with respect t B basis for M2x2. {1, x+1, x²+2x+1} and the standard

Find the matrix of the linear transformation T from R? R?, where T([1, 0]) = [1, -2], and T([2, 1]) = [2, 3] (Note: matrix A = [T([1,0]), T([0, 1])] ). 1 (a) 1 (b) 1. (c) -2 1 -2 3 -2 1 (d) 0. 3

Let T : R2 → R² be the linear transformation which rotates vectors counter-clockwise about the origin by 7T/3 radians (60 degrees). (i) Find the standard matrix of the linear transformation T. (ii) Use your matrix to rotate the vector v = counterclockwise about the origin by 4 T/3 radians.

Let T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector x = [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 0 0 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take 7 = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AȚ) is…