The Orthogonal Complement of a Unit Vector

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$, and suppose that $V$ is \reftext{def:finite-dimensional-vector-space-2026b}{finite-dimensional} and $V\ne\{0_{V}\}$; write $n=\dim V$ for its \reftext{def:dimension-inner-product-space-2026a}{dimension}.

Let $u\in V$ be a \reftext{def:unit-vector-2026a}{unit vector}, let $\tilde{u}\in V^{1}$ be the \reftext{def:finite-tuple-power-2026a}{$1$-tuple} with component $u$, and let $U=\operatorname{span}(\tilde{u})$ be its \reftext{def:span-finite-family-2026b}{span}, a \reftext{def:linear-subspace-2026a}{linear subspace} of $V$ by claim 1 of \ref{lem:span-is-subspace-2026b}. Let $W=U^{\perp}$ be the \reftext{def:orthogonal-complement-2026a}{orthogonal complement} of $U$, a linear subspace of $V$ by \ref{lem:orthogonal-complement-is-subspace-2026a}. Then the following hold.

\textbf{1. (Decomposition)} For every $x\in V$ there is exactly one pair consisting of a \reftext{def:complex-numbers-2026a}{complex number} $\lambda$ and a vector $w\in W$ with $x=\lambda u+w$; it satisfies $\lambda=\langle u,x\rangle$.

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

\textbf{3. (Splitting off a unit vector)} Suppose $n=r+1$ for some \reftext{def:natural-numbers-2026a}{natural number} $r$. Then $W\ne\{0_{V}\}$ and, with the inner product $W$ inherits by claim 3 of \ref{lem:subspace-inner-product-space-2026b}, there is an \reftext{def:orthonormal-basis-2026b}{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 \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} $[r]$ and $e_{r+1}=u$ is an orthonormal basis of $V$.