Is the following theorem true (bilinear form representation)

Question

I red a proof of the following statement: $f:V \times W \rightarrow F$ is a bilinear form, each of $B,B'$ is a basis of $V$ and each of $C,C'$ is a basis for $W$. $P$ is the transformation matrix from $B$ to $B'$ and $Q$ is the transformation matrix from $C$ to $C'$. $A$ represents $f$ with respect to $B,C$ and $A'$ represent $f$ with respect to $B',C'$. Then: $A' = P^tAQ$. I believe the statement needs to be the same but $P$ needs to be the transformation matrix from $B'$ to $B$ (which means $P \cdot [v]_B' = [v]_B$) and $Q$ needs to be the transformation matrix from $C'$ to $C$ ($Q \cdot [w]_C' = [w]_C$. Am I right?