1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) =
At. Compute the characteristic polynomial of T. You may wish to use
the basis of Mn×n(F) consisting of the matrices eij + eji, eij −eji
and eii.
2. Let A = (a b c d) (2 by 2 matrix) and let T :
M2×2(F) → M2×2(F) be defined asT (B) = AB. Represent T as a 4×4
matrix using the ordered basis {e11,e21,e12,e22}, and use this
matrix to...