6. The function S: M22 -> M22 defined as S(A) = AT - A is a linear transformation. a. Compute C = [S]B, where B is the standard basis of M22. b. Use Theorem 5.2.9 to lift bases of null C and col C back to bases for null S and im S. Theorem 5.2.9. Computing with matrix representations. Let T: V - W be linear, and let B and B' be ordered bases of V and W, respectively. The matrix A = [T]; satisfies the following properties: 1. v E nullT if and only if [v] B E null A; 2. WE im T if and only if [w] B E col A; 3. {v1, v2,. .., vr} is a basis of null T if and only if { [vi]B, [v2]B, . .., [V,]B} is a basis of null A; 4. {w1, w2, ..., ws} is a basis of im T if and only if {[wi]B, [w2]B, . .., [Vs]B'} is a basis of col A; 5. nullity T = nullity A and rank T = rank A. Proof. in-context c. Identify null S and im S as familiar subspaces of matrices