I need to determinate if this statement is true or false, some friends of mine think they found a proof but they are not sure meanwhile others seem to have a counterexample The statement is the following : Let $V$ a vector space of finite dimension on a field $K$, $W \subseteq V$ a subspace of $V$ and $f$ a nondegenerate symmetric bilinear form on $V \times V$. Then $V=W \oplus W^⊥$.
The statement is true for scalar products so I thought of a possible counterexample that can be the bilinear form given by the matrix $\begin{pmatrix} 1 & 0\\ 0& -1 \end{pmatrix}$ (that is not a scalar product) and $W=span\{(1,1)\}$ but the corollary 2.3 of this website seems to give a proof of the statement but with a restriction condition and I don't know if it can also be a proof for my statement https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2008-2009/bilinear-08.pdf.
Can someone help know if this statement is true or false ?