proof question:

2 2. Let A denote the 2 x 2 matrix We define a function (., .) : R2 x R2 - R by taking N (x, y) = y Ax, for each pair of vectors x, y E R2. (Technically, the result of computing y Ax is a 1 x 1 matrix, not a real number. However, we identify the 1 x 1 matrix with its single real entry.) (a) Does (., .) define an inner product on R2? If it does, prove it. If it does not, state which inner product conditions this function satisfies, and prove those. Also, provide counterexamples for each inner product condition that this function does not satisfy. (b) By replacing A with another matrix B different from A in only a single entry, define an inner product [., .] different from the function in part (a), and justify why this modified function is an inner product. (c) Give an orthonormal basis for R2 with respect to this newly-defined inner product [, ] that you chose in part (b). Justify that your chosen basis is orthonormal