Find an Inner Product of two polynomials

Question

Find an Inner Product function that has an ortonormal basis $B=\{1+2x,2+5x\}$ under vector space $R_2[x]$.

Well, I can tell that there are three requirements:

1. $\langle 1+2x,1+2x \rangle = 1$
2. $\langle 2+5x,2+5x \rangle = 1$
3. $\langle 1+2x,2+5x \rangle = 0$

but not sure how to use it in order to find such Inner Product.

Any ideas?

Answer 1

Hint

$$x=(2+5x)-2(1+2x) \\ 1=5(1+2x)-2(2+5x)$$

This allows you to write any $ax+b$ a linear combination of the two given polynomials.

You can then compute $$<ax+b, cx+d>$$

Hint 2 If $1+2x, 2+5x$ is a basis for your space, your space is the span of these two polynomials. Thus $V= \{ ax+b |a,b \}$.

Answer 2

We have that

$$1=5(1+2x)-2(2+5x)\implies \langle 1,1\rangle=25+4=29.$$

And

$$x=-2(1+2x)+(2+5x)\implies \langle x,x\rangle=4+1=5.$$

Finally

$$\langle 1,x\rangle=\langle 5(1+2x)-2(2+5x),-2(1+2x)+(2+5x)\rangle=-10-2=-12.$$