Trace and linear functional

Question

I was studying linear functionals and came across this question.

Suppose that for a given matrix $B$ in $M_n$, the linear space of $n\times n$ real matrices, a function $f_B:M_n\to \mathbb R$ is defined through $$f_B(A)=\mbox{trace}(B^t A)$$

Show:

1. $f_B$ is a linear functional.
2. Every linear functional is of the form $f_B$ for some $B$.

Answer 1

For 1, use the properties of the trace to prove that $$ f_B(c_1 A_1 + c_2 A_2) = c_1 f_b(A_1) + c_2 f_b(A_2) $$ For 2, note that the space of linear functionals over the $n^2$ dimensional space $M_n$ is $n^2$. Then, it suffices to find $n^2$ linearly independent functionals of the form $f_B$.