Given a symmetric, bilinear form $\phi: V \times V \rightarrow \mathbb{R}$ with $V$ having dimension $n$.
The map is given as $f_b(x)=x+\phi(x,b).b$ where $b \in V$, $b \neq 0$, $f_b: V \rightarrow V$.
This is a sample exam question that demands showing for eigenvalue $\lambda=1$ the dimension of the eigenspace is $n-1$.
I know that I can say $\phi(x,b)=x^T Ab$ but I fail to move on from here to show what is being demanded. Do I have to construct $A$ to show this?
Another idea I had was to show that $f$ has Jordan form and move from there but I am stuck.