Consider the function defined by: $$\langle p,q\rangle=2a_1b_1+a_2b_2$$ where $$p(x)=a_1+a_2x$$ and $$q(x)= b_1+b_2x $$
a) Verify that $$\langle p,q\rangle$$ defines an inner product on $\mathbb R.$
well thats the question, which ive kinda like a revision for my final exam soon. but im having a trouble solving it. i know i've like few rules to verify in order to prove that $$\langle p,q\rangle$$ defines an inner product. ive already solved 1st one $$\langle p,q\rangle=\langle q,p\rangle$$ $$\langle p,q\rangle=2a_1b_1+a_2b_2$$ $$=a_2b_2+2a_1b_1$$ $$=\langle q,p\rangle$$ which prove 1st rule$$ \langle p,q\rangle=\langle q,p\rangle $$ Not sure if its true ..
but im having a problem in the 2nd one where i should prove $$\langle p,q+z\rangle =\langle p,q\rangle+\langle p,z\rangle$$ ( where i define z and it should look like ) $$z(x)=c_1+c_2x$$
I'm having a problem which is i dont know to which part of $\langle p,q\rangle$ i should add the $z$
and if 1st rule verification is wrong.. inform me please thank you for your time
To verify that something is an "inner product" you have to show that it satisfies the definition of "inner product". And that is: 1) is linear in the first variable. That is, that = a+ b.
2) That is "symmetric" if the field is the real numbers: = or "conjugate symmetric" if the field is the complex numbers: = the complex conjugate of .
3) That is a positive real number for every v.
1) With $u= a_1+ b_1X$, $v= a_2+ b_2X$, and $w= a_3+ b_3X, $u+ v= (a_1+ a_2)+ (b_1+ b_2)X$ so that $= 2(a_1+ a_2)(b_1+ b_2)+ a_3b_3$. $= 2a_1b_1+ a_3b_3$ and $= 2a_2b_2+ a_3b_3$. $+ = 2a_1b_1+ 2a_2b_2+ 2a_3b_3$. NO, that is not the same as . This is NOT an inner product!
Are you sure you copied the problem correctly? With $u= a_1+ b_1X$ and $v= a_2+ b_2X$, $<u, v>= 2a_1a_2+ b_1b_2$ would make more sense.