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 $V$.

just imagine is as a n dimensional space. w is one dimension. except it there remains n dimension which are orthogonal/perpendicular to w, which is exactly what P is , set of vectors orthogonal to w.
