What does it mean when you say a matrix of of a linear transformation is relative to a standard basis?

Question

I'm supposed to show that the matrix A^t relative to the B = {e1, e2, ..., en} for V is the usual matrix transpose of (aij). I'm just having difficulties understanding what it means. We were given the hint to express A^t(ei) in terms of the basis B using < A(x), y > = < x, A^t(y) >, and this is the dot product of the entries.

Answer 1

If your teacher hasn't explained those terms anywhere, they are a terrible teacher.

That being said, the "standard basis" is typically $e_1=(1,0,0,0,\cdots)$, $e_2=(0,1,0,0,\cdots)$ and so on. Expressing a matrix $A$ relative to a basis whose basis vectors are the columns of matrix $B$ is done using this formula $$A_{\text{in new basis}}=B A B^{-1}$$

Answer 2

At the beginning, you must be aware of the fact that you can constant different matrices for the same linear operator by choosing different bases. And you should know how to contact them. Now, the answer to your question is met by choosing the standard basis, which is $(e_1, e_2,\dots, e_n)$.