Let $T: V \rightarrow V$ be a self adjoint linear map where $V$ is an inner product space.
Known fact: With respect to any orthonormal basis- the the matrix for $T$ is conjugate symmetric. When dealing with a real vectorspace just symmetric.
However is it true that if there exists some basis (not necessarily orthonormal) such that the matrix for $T$ is symmetric then $T$ is self adjoint?