## 每週問題 June 12, 2017

Let $\mathbf{v}_1,\ldots,\mathbf{v}_n$ be a basis of an inner product space. Prove that there exists an orthogonal basis $\mathbf{e}_1,\ldots,\mathbf{e}_n$ such that $\mathbf{e}_i\in\hbox{span}\{\mathbf{v}_1,\ldots,\mathbf{v}_i\}$ for all $i=1,\ldots,n$.

$\mathbf{e}_{k+1}=c_1\mathbf{e}_1+\cdots+c_k\mathbf{e}_k+\mathbf{v}_{k+1}$

$\displaystyle c_i=-\frac{\left\langle\mathbf{e}_i,\mathbf{v}_{k+1}\right\rangle} {\left\langle\mathbf{e}_i,\mathbf{e}_i\right\rangle},~~~ i=1,\ldots,k$