Proposition (Gran-Schmidt process (212))

Let $V$ be an inner product space and suppose that $v_{1},v_{2},\dots,v_{m}$ are linearly independent in $V$. We define $u_{1},u_{2},\dots,u_{m}$ inductively as: $$u_{1}=\frac{v_{1}}{\lVert v_{1} \rVert }$$and $$u_{k}= \frac{v_{k}-\langle v_{k}, u_{1} \rangle u_{1}\langle v_{k}, u_{2} \rangle u_{2}-\dots-\langle v_{k}, u_{k-1} \rangle u_{k-1}}{\lVert v_{k}-\langle v_{k}, u_{1} \rangle u_{1}\langle v_{k}, u_{2} \rangle u_{2}-\dots-\langle v_{k}, u_{k-1} \rangle u_{k-1} \rVert }.$$ The set $\{ u_{1},u_{2},\dots,u_{m} \}$ is an orthonormal set of vectors such that $$\mathrm{span}(u_{1},u_{2},\dots,u_{j})=\mathrm{span}(v_{1},v_{2},\dots,v_{j}),\quad j=1,2,\dots,m$$