Let the vector space $P_2$ have the inner product:
$\langle p,q\rangle=\int\limits_{-1}^{1}p(x)q(x)dx$
Apply the Gram-Schmidt process to transform the standard $S=\{1,x,x^2\}$ into an orthonormal basis.
The book does not provide solution for this problem and I do not know how to solve it.
1-How can I transform the S basis into an orthonormal basis if I need at least three vectors since the dimension of the vector space in cause is 3?
2-Can someone provide me a solution?
Thanks in advance!
$\endgroup$ 11 Answer
$\begingroup$First, normalize first vector of the basis $\;v_1=1\;$:
$$\langle v_1,v_1\rangle=\langle 1,1\rangle:=\int_{-1}^11\cdot dx=2\implies \color{red}{u_1=\frac{v_1}{\left\|v_1\right\|}=\frac1{\sqrt2}}$$
Next, orthogonalize second vector wrt the first one:
$$w_2:=v_2-\langle v_2,u_1\rangle u_1=x-\left\langle x,\frac1{\sqrt2}\right\rangle \frac1{\sqrt2}=x-\frac12\int_{-1}^1x\,dx=x-\left.\frac14x^2\right|_{-1}^1=x$$
Now, orthonormalize that last vector:
$$\langle x,x\rangle=\int_{-1}^1x^2dx=\left.\frac13x^3\right|_{-1}^1=\frac23\implies\color{red}{u_2=\frac{w_2}{\left\|w_2\right\|}}=\sqrt\frac32\,x$$
Last step, and this you will do: orthogonalize third vector wrt the first two:
$$w_3:=x^2-\langle x^2,u_1\rangle u_1-\langle x^2,u_2\rangle u_2$$
and then take
$$\color{red}{u_3=\frac{w_3}{\left\|w_3\right\|}}$$
$\endgroup$ 4
