## 每週問題 December 17, 2012

Let $C[0,1]$ be the space of continuous functions on the interval $[0,1]$ with the inner product defined by

$\displaystyle\left\langle f,g\right\rangle=\int_0^1f(x)g(x)dx$.

Let $\mathcal{P}_2=\mathrm{span}\{1,x,x^2\}$. Apply the Gram-Schmidt algorithm to the basis $1, x, x^2$ to obtain an orthogonal basis for $\mathcal{P}_2$.

$\mathbf{u}_1=1$$\mathbf{u}_2=x$$\mathbf{u}_3=x^2$。Gram-Schmidt 正交化程序如下：

\begin{aligned} \mathbf{v}_1&=\mathbf{u}_1=1\\ \mathbf{v}_2&=\mathbf{u}_2- \frac{\left\langle\mathbf{v}_1,\mathbf{u}_2\right\rangle} {\left\langle\mathbf{v}_1,\mathbf{v}_1\right\rangle}\mathbf{v}_1=x-\frac{1/2}{1}1= x-\frac{1}{2}\\ \mathbf{v}_3&=\mathbf{u}_3- \frac{\left\langle\mathbf{v}_1,\mathbf{u}_3\right\rangle} {\left\langle\mathbf{v}_1,\mathbf{v}_1\right\rangle}\mathbf{v}_1- \frac{\left\langle\mathbf{v}_2,\mathbf{u}_3\right\rangle} {\left\langle\mathbf{v}_2,\mathbf{v}_2\right\rangle}\mathbf{v}_2\\ &=x^2-\frac{1/3}{1}1-\frac{1/12}{1/12}\left(x-\frac{1}{2}\right)=x^2-x+\frac{1}{6}.\end{aligned}

\begin{aligned} \mathbf{q}_1&=\frac{1}{\sqrt{\left\langle \mathbf{v}_1,\mathbf{v}_1\right\rangle}}\mathbf{v}_1=\frac{1}{1}\mathbf{v}_1=1\\ \mathbf{q}_2&=\frac{1}{\sqrt{\left\langle \mathbf{v}_2,\mathbf{v}_2\right\rangle}} \mathbf{v}_2=\frac{1}{\sqrt{1/12}}\left(x-\frac{1}{2}\right)=2\sqrt{3} \left(x-\frac{1}{2}\right)\\ \mathbf{q}_3&=\frac{1}{\sqrt{\left\langle \mathbf{v}_3,\mathbf{v}_3\right\rangle}} \mathbf{v}_3=\frac{1}{\sqrt{1/180}}\left(x^2-x+\frac{1}{6}\right)=6\sqrt{5} \left(x^2-x+\frac{1}{6}\right).\end{aligned}

PowSol-Dec-17-12