Can someone explain this example ? (Gram-schmidt Orthago

[sid9221]

http://dl.dropbox.com/u/33103477/4.png

Can someone explain to me how the bits in red are calculated with Integration, the examples are doing my head in and it would be really useful if a human could show me how they are getting the values cause mine are different.

 

[sunjin09]

For example, the first circle is the integration of t times 1, from 0 to 1, which is equal to $\frac{t^2}{2}|_0^1=\frac{1}{2}$. The rest are the same

 

[vela]

Here's a bit more detail:
\begin{align*}
\langle 1, 1 \rangle &= \int_0^1 1\cdot 1 \, dt = t\big|_0^1 = 1 \\
\langle t, 1 \rangle &= \int_0^1 t\cdot 1 \, dt = \left. \frac{t^2}{2}\right|_0^1 = \frac{1}{2}
\end{align*} Those are, respectively, the denominator and numerator in the first term you circled.