I have a vector space $V$ such that $V = A\oplus A^\perp$ i.e. $V$ is a direct sum of its subspace $A$ and orthogonal complement of $A$. Suppose we also have $V = A \oplus C$
Then can we say that $A^\perp = C$. If not then in what condition this relation may hold true? I think both subspace will have same dimension but i am not sure about equality of sub spaces.
I am confused here and need a clarification.