1. ## Inner product/Projections

Use the inner product

in to find the orthogonal
projection of onto the line spanned by .

.

Which is wrong.

The projection of f onto L is given by f(x)-{g(x) dot f(x)/||g(x)||^2}*g(x)

The inner product of g(x) and f(x) is
(3 (-3)^2+6 (-3)+3) (2 (-3)^2-6 (-3)-9)+(3(0)^2 +6(0) +3)*(2(0)^2 -6(0) -9)+(3(3)^2 +6(3) +3)*(2(3)^2 -6(3) -9)= -135

Inner product of g(x) with itself is then:
(2(-3)^2 -6(-3) -9)*(2(-3)^2 -6(-3) -9)+(2(0)^2 -6(0) -9)*(2(0)^2 -6(0) -9)+(2(3)^2 -6(3) -9)*(2(3)^2 -6(3) -9) = 891

-135/891 simplifies to -5/33

Multiplying g(x) with -5/33 gives -(10 x^2)/33+(10 x)/11+15/11

Subtracting this value from f(x) results with
(109 x^2)/33+(56 x)/11+18/11, which is my final answer.

2. I'm also having difficulty with this question.

Use the inner product in the vector space to find the orthogonal projection of onto the subspace spanned by and .
(Caution: and do not form an orthogonal basis of .)

I think I am going wrong because of minor miscalculations? My answer for this question is 5 x^2-5 x+5/6, which is incorrect.