Dimension of a subspace of finite-dimensional product space $V$ equals $\dim V - 1$ Suppose $w$ is a nonzero vector in a finite dimensional inner product space $V$. Let $P = \{ v \in V | \langle v,w\rangle = 0\}$. Show that $\dim P = \dim V - 1$ where $P$ is a subspace of $V$.

Let $w=(a_1,\ldots, a_n), v=(x_1,\ldots, x_n)$. Then $P$ is defined by one equation $a_1x_1+\ldots +a_nx_n=0$, hence $\dim P=\dim V-1$.
