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

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.

• The relationship between the dimension of the kernel of a matrix product and the dimensions of the individual kernels seems applicable.
– amd
Commented Dec 2, 2017 at 23:06

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.