Problem: Let $V$ be the real vector space of the real continuous functions defined on $[0,1]$ with the inner product given by
\[
\left\langle f,g \right\rangle = \int _0^1 f(x) g(x) \ \mathrm{d} x.
\]
Prove that the set $S = \left\{ 1, x, x^2, x^3 \right\} $ is linearly independent and obtain an orthonormal set of vectors whose span is same as that of the set $S$.
Solution: Note that the Wronskian of these functions are
\begin{align*}
W =
\det \begin{bmatrix}
1 & x & x^2 & x^3 \\
0 & 1 & 2x & 3x^2 \\
0 & 0 & 2 & 6x \\
0 & 0 & 0 & 6 \\
\end{bmatrix} = 12 \neq 0.
\end{align*}
Thus, the set $S$ is linearly independent.
We now need to apply the Gram-Schmidt process to obtain an orthogonal set whose span is same as that of $S$. Let us write $u_1 = 1$, $u_2 = x$, $u_3 = x^2$ and $u_4 = x^3$.
-
$v_1 = u_1 = 1$.
-
Obtaining the second vector $e_2$:
\begin{align*}
v_2 = u_2 - \frac{\left\langle u_2 , v_1 \right\rangle }{\left\langle u_1 , u_1 \right\rangle } u_2
\end{align*}
Will be upated soon