Applying properties of an inner product to prove inner products.

Question

I was reading a chapter about inner products in a book on linear algebra and came across this question:

"a) Consider the vector space $P_2(\mathbb{R})$ of real polynomials of degree $\leq2$.
Which of the following forms of $P_2$ have inner products?

(1) $<p,q>=p(0)q(0)$
(2) $<p,q>=p(0)q(0)+p'(0)q'(0)+p''(0)q''(0)$

b) Calculate $<t+1, t^2+1>$ for each of the inner products in a)."

I know that these 4 properties have to be satisfied to prove it is an inner product: $$1) \langle u+v,w \rangle=\langle u,w\rangle+\langle v,w\rangle$$ $$2) \langle\alpha v, w\rangle=\alpha \langle v,w\rangle$$ $$3) \langle v,w\rangle =\langle w,v\rangle$$ $$4) \langle v,v\rangle\geq 0$$

However I am not completely sure on how to apply the definition to my actual problem. And also how do I apply b) to this?

Answer 1

$a)$: Each form is an inner product.

$b)$: Take $p(t) = t+1, q(t) = t^2+1 \implies p(0) = 1, p'(0) = 1, p''(0) = 0, q(0) = 1, q'(0) = 0, q''(0) = 2 \implies$

• $<p,q>_1 = p(0)q(0) = 1\cdot 1 = 1$
• $<p,q>_2 = 1\cdot 1 + 1\cdot 0 + 0\cdot 2 = 1$