Consider P2 with the inner product given by evaluation at -1, 0, and 1. That is < p, q >= p(−1)q(−1) + p(0)q(0) + p(1)q(1). Let p(t) = 3t − t2 and q(t) = 3 + 2t2 .
A) Compute < p, q >, ||p||, and ||q||.
B) Compute the orthogonal projection of q onto the subspace spanned by p.

A).

B).The orthogonal projection of q onto the subspace spanned by p is

We have

And < p, q >= p(−1)q(−1) + p(0)q(0) + p(1)q(1).

Now,

A).

B)Let V be the subspace spanned by p.

Now, an orthonormal basis for p is

 Then the projection of q onto the subspace spanned by p is

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(A)

Evaluate and at the given values of , then plug them into the inner product:

Now,

(B)

Let denote the subspace spanned by . We need an orthonormal basis for . Since

we have the basis vectors ; normalize these vectors to get the orthonormal basis,

Then the projection of onto is

idk what the heck he answer is, i’m just putting this here so i finish my account