Gram–Schmidt process

《Algebraic and Analytic Geometry》 · 第 344 页 · Section 6.2「Gram–Schmidt Orthogonalization Process and Orthogonal Complements」，Theorem 6.4

Let $V$ be an inner product space and $S=\{w_1,\ldots,w_n\}$ be a linearly independent subset of $V$. Define $S'=\{v_1,\ldots,v_n\}$, where $v_1=w_1$ and $v_k=w_k-\sum_{j=1}^{k-1}\frac{\langle w_k,v_j\rangle}{\|v_j\|^2}v_j$ for $2\le k\le n$; then $S'$ is an orthogonal set of nonzero vectors such that $\operatorname{span}(S')=\operatorname{span}(S)$. The construction of $\{v_1,v_2,\ldots,v_n\}$ by the use of Theorem 6.4 is called the Gram–Schmidt process.