1. Definitions. State precisely the following definitions. One point each. (a) A transformation T is a linear transformation. b) A mapping T : R" - Rm is onto. (c) A mapping T : R" - R" is one-to-one. (d) If A is an m x n matrix and B is an n x p matrix with columns b1, ..., bp, define the product AB. (e) An n x n matrix A is invertible. 2. True False. Mark each as T for "True" or F for "False". One point each. (a) For an n x n matrix A, if A2 = 0 then A = 0. ( b ) If A is an n x n matrix, then the equation Ax = b has at least one solution for each b in ". (c) If the columns of A are linearly dependent, then det A = 0. (d) If A and B are both m x n, then both AB and AB are defined. (e) If A is an n x n matrix, then det(A-!) = -det(A). 3. Suppose that A is an n x n invertible matrix. Explain why A'A is invertible. Then show that A-1 = (AT A)-1AT. 4. Given the linear transformation T : R2 - R4 defined by T(X1, X2) = (2, 21, 3X1 - 22, -201 + 12) answer the following parts: (a) Find the matrix A so that T(x) = Ax. (b) Is T one-to-one? Why? (c) Is T onto? Why