- #1
Mr Davis 97
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Homework Statement
Find an orthogonal basis for ##\operatorname{span} (S)## if ##S= \{1,x,x^2 \}##, and ##\langle f,g \rangle = \int_0^1 f(x) g(x) \, dx##
Homework Equations
The Attempt at a Solution
So we start by the normal procedure.
Let ##v_1 = 1##. Then ##\displaystyle v_2 = x - \frac{\langle x,1 \rangle}{\| 1 \|^2}(1) = x - \frac{1}{2}##.
Then ##\displaystyle v_3 = x^2 - \frac{\langle x^2,1 \rangle}{\| 1 \|^2}(1) - \frac{\langle x^2,x \rangle}{\| x \|^2}(x) = x^2 - \frac{1}{3} - \frac{3}{4}x##.
But this is not correct, because if I calculate ##\displaystyle \langle 1, x^2 - \frac{1}{3} - \frac{3}{4}x\rangle = \int_0^1 x^2 - \frac{1}{3} - \frac{3}{4}x \, dx = -\frac{3}{2} \ne 0##.
What am I doing wrong?