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 $V$.

Since $P$ is the kernel of the linear mapping $v\mapsto \langle v,w\rangle$, it can be described (in any coordinates) as a nullspace of the matrix $w^T$. Now use what you know about matrix row reduction to finish the problem.
