The Orthogonal Complement of a Unit Vector

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$, and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$; write $n=\dim V$ for its dimension.

Let $u\in V$ be a unit vector, let $\tilde{u}\in V^{1}$ be the $1$-tuple with component $u$, and let $U=\operatorname{span}(\tilde{u})$ be its span, a linear subspace of $V$ by claim 1 of The Span of a Finite Family is the Smallest Subspace Containing It. Let $W=U^{\perp}$ be the orthogonal complement of $U$, a linear subspace of $V$ by The Orthogonal Complement of a Linear Subspace is a Linear Subspace. Then the following hold.

1. (Decomposition) For every $x\in V$ there is exactly one pair consisting of a complex number $\lambda$ and a vector $w\in W$ with $x=\lambda u+w$; it satisfies $\lambda=\langle u,x\rangle$.

2. (Dimension one) If $n=1$, then $W=\{0_{V}\}$.

3. (Splitting off a unit vector) Suppose $n=r+1$ for some natural number $r$. Then $W\ne\{0_{V}\}$ and, with the inner product $W$ inherits by claim 3 of A Linear Subspace is a Vector Space and Inherits an Inner Product, there is an orthonormal basis $f\in W^{r}$ of $W$. Moreover the tuple $e\in V^{r+1}$ with $e_{k}=f_{k}$ for $k$ in the initial segment $[r]$ and $e_{r+1}=u$ is an orthonormal basis of $V$.