Dimension of the orthogonal complement of subspace in a degenerated bilinear form

Question

Let g be a bilinear form (not necessarily a non-degenerate one) of a space $\mathbf V$, such that $dim \mathbf V = n$ and g: $\mathbf V \times \mathbf V \rightarrow \mathbf V$. Consider a subspace $\mathbf W$ of $\mathbf V$, such that $dim \mathbf W = m$. Prove that $dim \mathbf W ^\bot \geq n-m$.

Ps.: I know that for a bilinear form non degenerate holds $dim \mathbf W ^\bot = n-m$, I'm having trouble proving the inequality when the form is degenerate.

Answer 1

I will assume that you are talking about a symmetric bilinear form.

The bilinear form $g$ yields a linear application from $V$ to its dual, which maps $v$ to $g(v,-)$.

We have that the locus of zeros of the image of $W$ through this application is the orthogonal space to $W$, and its dimension is equal to the codimension of the image of $W$. Hence the inequality.