# how to find inner product and linear transformation

1.) $p=p(x)$ and $q=q(x)$ are polinom in $P_2$. Defined by $<p,q>=p(0)q(a)+p(\frac{1}{2})$$q(b)+p(1)q(c) for certain a,b,c such that the <.,.> is inner product inP_2 . Find the value of \left\lVert 4x^2-1\right\rVert I can not determine the value of a,b,c such that <,> is inner product 2.) T_1 : R^(3x3) -> R and T_2 : R^(3x3)->R^(3x3) are linear transformation defined by T_2(A)=A^T. If A=\begin{matrix} a & b & c \\d & e & f \\g & h & i \\ \end{matrix} and (T_1(T_2(A))=T_1(A) , find T_1(A) T_1(A)=T_1(A^T) using this fact how can we do for solving this question • @DonAntonio can you give the absolute value for ∥4x^2-1∥ ? or just in variable a,b,c? – lio Oct 30 '16 at 11:20 • This site is not for dumping undigested homework questions. Tell us what you know about these questions, please! Also, what is "Jika"? – Gerry Myerson Oct 30 '16 at 11:55 • @lio Reading what you wrote, only as function of \;a,b,c\; ...and that is not "absolute value", though I guess one could think of if that way. Its name is norm . – DonAntonio Oct 30 '16 at 12:34 • @DonAntonio i think find ∥ 4x^2-1∥ for exact real number, which not using a,b,c as variable in the value of ∥ 4x^2-1∥ – lio Oct 30 '16 at 14:19 • Then you have to do much, but really much more work than what you asked. You must first find \;a,b,c\; such that the given function is really an inner product...and then calculate that element's norm. – DonAntonio Oct 30 '16 at 15:23 ## 1 Answer$$\left\|4x^2-1\right\|^2=\langle 4x^2-1,\,4x^2-1\rangle=4\langle x^2,\,x^2\rangle-8\langle x^2,\,1\rangle+\langle 1,\,1\rangle$$and now apply the definition. For example, taking \;q(x):=x^2\; , we get$$\langle x^2,\,x^2\rangle=q(0)q(a)+q\left(\frac12\right)q(b)+q(1)q(c)=\frac{b^2}4+c^2$$and etc. • I think part of the point is that$a,b,c$have to be chosen carefully to make the thing an inner product. – Gerry Myerson Oct 30 '16 at 11:53 • @GerryMyerson I would also think that would be a nice exercise, yet that is not what is asked in the question...Perhaps they just want the value of$\;\left\|4x^2-1\right\|\;$as a function of$\;a,b,c\;\$ . – DonAntonio Oct 30 '16 at 12:33
• @GerryMyerson I already edit the question.I misinterpreted the question maybe. can you help me to answer this question? – lio Oct 30 '16 at 14:26