Induced Bilinear Form on $V/A$

Question

We are given a matrix

\begin{bmatrix} 2&3&3&0\\ 3&4&3&-1\\ 3&3&0&-3\\ 0&-1&-3&-2 \end{bmatrix}

and we want to find the induced bilinear form on $V/A$ where

$A = span\{3e_1-2e_2+e_4, 3e_1-3e_2+e_3\}$

I'm not really sure where to start with this question. We think we should find a basis for $V/A$, but not really sure after that.

Any help would be appreciated.

Answer 1

Let $M$ denote the matrix, and $u,v$ the given spanning vectors of $A$.

The main thing is that $Mu=Mv=0$, so that $Ma=0$ for every $a\in A$, and thus the bilinear form $$(x+A,\, y+A) \mapsto x^TMy$$ is well defined: we get the same value for $x'=x+a$ and for $y'=y+a'$, with arbitrary $a, a'\in A$.

To obtain a $2\times 2$ matrix, just fix a basis $b_1,b_2$ of $V/A$ and put $b_i^TMb_j$ in the $i,j$ entry.