2021-35.md

course	course_year	question_number	tags	title	year
Linear Algebra	IB	35	IB 2021 Linear Algebra	Paper 1, Section I, $1 \mathrm{E}$	2021

Let $V$ be a vector space over $\mathbb{R}, \operatorname{dim} V=n$, and let $\langle,$,$rangle be a non-degenerate anti-$ symmetric bilinear form on $V$.

Let $v \in V, v \neq 0$. Show that $v^{\perp}$ is of dimension $n-1$ and $v \in v^{\perp}$. Show that if $W \subseteq v^{\perp}$ is a subspace with $W \oplus \mathbb{R} v=v^{\perp}$, then the restriction of $\langle,$,$rangle to W$ is nondegenerate.

Conclude that the dimension of $V$ is even.