I need some help with this.
So I had question at my exam that goes like this: Let $P_{\leq 2}$ be a vector space of all polynomials. Check if scalar(inner) product is correct:
$\langle p,q \rangle=p(1)\times q(1)+2p(0)\times q(0)+p(-1)\times q(-1)$
b) find $||x^2-3x+1||$
So I studied that to check if this is correct scalar product I need to check this:
1.$\langle x,x \rangle\geq 0$
$\langle x,x \rangle = 0\Leftrightarrow x=0$
2.$\langle x,y \rangle = \langle y,x \rangle$
3.$\langle \lambda x,y \rangle = \lambda\langle x,y \rangle$, $\lambda \in R$
4.$\langle x+y,z \rangle = \langle x,z \rangle + \langle y,z \rangle$
NOTE: Don't get confused with using $x,y,z$ from above, it's not related to anything from question.
So we have that $p(x)= a_{0}+a_{1}x+a_{2}x^2$, and $q(x)= b_{0}+b_{1}x+b_{2}x^2$
Now I should find $p(1),q(1)…$ and replace it ?
Am I doing this right or just wasting time trying to proof that 4 properties from above?
For the second part(b part) neither do I understand is it related for the first part(a part), for the norm I calculated like this for example:
$x=(1,2,3)$
$||x||= \sqrt{1^2+2^2+3^2}$ but from the example from above I don't know how to solve it, because I'am pretty sure that I can't just calculate $||x^2-3x+1||$ as a $||x^2||-||3x||+||1||$ This should't make any sense.
And sorry, had to translate everything from native language, thanks for help
Best Answer
Yes, to prove that $\left \langle ,\right \rangle$ defines an inner product on $P_{\le 2}$ (I assume you mean $P_{\le 2}$ is set of polynomials with degree at most $2$), you need to prove the above-mentioned four properties for any $x,y \in P_{\le2}$. To achieve this, your idea of letting $p(x)=a_0+a_1x+a_2x^2,q(x)=b_0+b_1+b_2x^2$ and trying to find $p(1),q(1),p(0),$ etc. is on the right track. Keep going.
That is not the definition of norm with respect to the inner product $\left \langle, \right\rangle$ defined in the begining. It should be $\|x\|= \sqrt{\left \langle x,x \right \rangle}$.