| course |
course_year |
question_number |
tags |
title |
year |
Linear Algebra |
IB |
40 |
|
Paper 4, Section II, F |
2012 |
Let $V$ be a finite-dimensional real vector space of dimension $n$. A bilinear form $B: V \times V \rightarrow \mathbb{R}$ is nondegenerate if for all $\mathbf{v} \neq 0$ in $V$, there is some $\mathbf{w} \in V$ with $B(\mathbf{v}, \mathbf{w}) \neq 0$. For $\mathbf{v} \in V$, define $\langle\mathbf{v}\rangle^{\perp}={\mathbf{w} \in V \mid B(\mathbf{v}, \mathbf{w})=0}$. Assuming $B$ is nondegenerate, show that $V=\langle\mathbf{v}\rangle \oplus\langle\mathbf{v}\rangle^{\perp}$ whenever $B(\mathbf{v}, \mathbf{v}) \neq 0$.
Suppose that $B$ is a nondegenerate, symmetric bilinear form on $V$. Prove that there is a basis $\left{\mathbf{v}{1}, \ldots, \mathbf{v}{n}\right}$ of $V$ with $B\left(\mathbf{v}{i}, \mathbf{v}{j}\right)=0$ for $i \neq j$. [If you use the fact that symmetric matrices are diagonalizable, you must prove it.]
Define the signature of a quadratic form. Explain how to determine the signature of the quadratic form associated to $B$ from the basis you constructed above.
A linear subspace $V^{\prime} \subset V$ is said to be isotropic if $B(\mathbf{v}, \mathbf{w})=0$ for all $\mathbf{v}, \mathbf{w} \in V^{\prime}$. Show that if $B$ is nondegenerate, the maximal dimension of an isotropic subspace of $V$ is $(n-|\sigma|) / 2$, where $\sigma$ is the signature of the quadratic form associated to $B$.