Problem:
Let $F:V \times W \to \mathbb{R}$ be a non degenerate bilinear form. The question is: prove that $V$ and $W$ have the same dimension (the vector spaces $V$ and $W$ are finite dimensional)
My answer is: $F$ is non degenerate, then the matrix of $F$ is invertible, which means it's square and this implies that $V$ and $W$ have the same dimension. Is my assumption that the matrix of a non degenerate bilinear form is invertible true? Also let me know if my answer is true?