Problem 1. Assume that a is any scalar, and that A, B, and C are matrices of sizes such that the indicated matrix products are defined. Prove the following:1

(i) (associativity) A(BC) = (AB)C.

(ii) (distributivity) A(B + C) = AB + AC.

(iii) (scalar commutativity) aA = Aa.

(iv) (AB) T = BT AT .

Problem 2. Let A be an m n-matrix and let ~u = ha1, . . . , ani = a1 . . . an , and ~v = hb1, . . . , bni = b1 . . . bn be two vectors in R n . Prove the following:

(i) A(~u + ~v) = A~u + A~v

(ii) A(c~u) = c(A~u), for any scalar c.

Problem 3. Draw the vector ~v = h1, 0i, and then draw the vectors cos sin sin cos ~v = cos sin sin cos 1 0 when = 0, /4, /2, , and 3/2.

Problem 4. Consider the two square matrices Ann and Bnn. Suppose that A and B commute, meaning that AB = BA. Show that (A B)(A + B) = A 2 B 2

Problem 5. A matrix A is called symmetric if A = AT . Note that, a symmetric matrix A is necessarily a square matrix (if A is m n, then AT is n m, so A = AT forces n = m). If A is any matrix, show that the matrix AAT is symmetric. Hint: you can use (iv) of Problem 1. (

Problem 6. Let T : R 2 R 2 be a linear transformation. In each case, find the the transform matrix T.

(i) T is a reflection about the y-axis.

(ii) T is a reflection about the line y = x.

(iii) T is a reflection about the line y = x.