For $s(t) = a_2t^2 + a_1t + a_0$ and $r(t) = b_2t^2 + b_1t + b_0$ in $P_2[t],$ define $<s(t),r(t)> = 2a_2b_2 + a_1b_1 + 3a_0b_0$

Question

EDIT: I'm still trying to figure out! Will ask for help if I can't answer it still by the end of the day. Thank you :)

Online classes hasn't been as easy as face-to-face classes and with that my professor is quite old and isn't very familiar with the online methods of teaching these days.

He said that the above is an inner product on $P_2[t].$ In attempting to prove other < , > examples, I would like to know how to know and show that < , > are inner products on $P_2[t].$ You may use the example I provided. Thank you so much!

Answer 1

an orthonormal basis is given by $$ \frac{t^2}{\sqrt 2}, \; \; \; t, \; \; \; \frac{1}{\sqrt 3} $$

If you have real numbers with $$e^2 + f^2 + g^2 = 1,$$ a unit vector is given by $$ \frac{ e t^2}{\sqrt 2} \; \; \; +ft + \; \; \; \frac{g}{\sqrt 3} $$

The quadratic form is given by the norm of your $s(t)$ being $$ 3 a_0^2 + a_1^2 + 2 a_2^2 $$ which becomes $g^2 + f^2 + e^2 $ when $a_0 = g/ \sqrt 3, \; \; a_1 = f, \; \; a_2 = e/ \sqrt 2$