Chapter 4.2, Problem 33E

Let M_2×2 be the vector space of all 2 × 2 matrices, and define T : M_2×2 → M_2×2 by T(A) = A + A^T, where

$\text{A}=\left[\begin{array}{cc}\text{a}& \text{b}\\ \text{c}& \text{d}\end{array}\right]$.

1. a. Show that T is a linear transformation.
2. b. Let B be any element of M_2×2 such that B^T = B. Find an A in M_2×2 such that T(A) = B.
3. c. Show that the range of T is the set of B in M_2×2 with the property that B^T = B.
4. d. Describe the kernel of T.