Please explain 11-15

(11) Which of the following functions are linear? Why? (a) The function f: R2 -> R2 defined by f ( = 2 x ty - 1 -2 - 4y + 1 (b) The function g: R2 - R2 defined by " ( [ ] ) - [Gu - 22]. (12) Which of the following functions are linear? Why? (a) The function f : R[x] - R[x] which, given a polynomial (x), returns the polynomial (22+ 2)4(x). (b) The function g: R[x] -> R2 which, given a polynomial ana" + an-12 1+ . . . +a2x2 + aix tao, returns the vector (13) For any positive integers p and q, explain why it always possible to find a matrix A such that p = dim(Nul(A) ) and q = dim(Nul(AT) ). (14) Let A be an m x n matrix and f: R" - Rm be the function f(x) = Ax. Three of the following statements are equivalent to each other, and one of them is not equivalent to the rest; which one is not equivalent to the others? Explain why. (a) The function f is onto. (b) dim Row(A) = m. A has more columns than rows. Col(A) = Rm. (15) Let A be an m x n matrix and f: R" -> Rm be the function f(x) = Ax. Three of the following statements are equivalent to each other, and one of them is not equivalent to the rest; which one is not equivalent to the others? Explain why. (a) Ax = 0 has exactly one solution. (b) The function f is one-to-one. (c) The columns of A are linearly independent. d) A has a pivot in each row. sets of R2 are subspaces? In each case, provide an explanation