Here's what I would have submitted:


Question 1:

This is the solution to A^t A x = A^t b, so
[12  8]  [-24]
[ 8 10]x=[ -2],

  [-4]
x=[ 3]



Question 2:

Plugging in x=0, 1, and 2, respectively, we get:
a+0b+0c=0
a+b+c=1
a+2b+4c=5.

These can be solved to yield the identity
1+4+...+x^2=x(x+1)(2x+1)/6



Question 3:

We have (sin x, cos x) = (integral from 0 to 2pi) 1/2 sin 2x = 0, so
the basis
{sin x, cos x} is already orthogonal.

This and similar facts are the foundation of the theory of Fourier
series and Fourier transforms, which we'll see next week.