Let V = R^2×2 be the vector space of 2-by-2 matrices with realentries over
the scalar field R. We can define a function L on V by
L : V is sent to V
L = A maps to A^T ,
so that L is the “transpose operator.” The inner product of twomatrices B in R^n×n and C in R^n×n is usually defined to be
:= trace (BC^T) ,
and we will use this as our inner product on V . Thus when we talkabout
elements B,C in V being orthogonal, it means that :=trace (BC^T) = 0.
Problem 1.
1. First show that L is linear, so that L in B (V ).
2. Now choose a basis for the vector space V = R^2×2, and find thematrix of
L with respect to your basis.
