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

you have 6 cupcakes

walter needs to make at least 5 more cookies but no more than 40. 5 ≥ x ≥ 40;

step-by-step explanation:

walter has to make a minimum of 20 cookies. he already has 15 cookies made, so he must make 5 more to meet the minimum.

the maximum amount of cookies he can make is 55. he already has 15, so he has to make no more than 40 more cookies to meet the maximum.