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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 in$P_2$ . 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

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  • $\begingroup$ @DonAntonio can you give the absolute value for ∥4x^2-1∥ ? or just in variable a,b,c? $\endgroup$ – lio Oct 30 '16 at 11:20
  • $\begingroup$ This site is not for dumping undigested homework questions. Tell us what you know about these questions, please! Also, what is "Jika"? $\endgroup$ – Gerry Myerson Oct 30 '16 at 11:55
  • $\begingroup$ @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 . $\endgroup$ – DonAntonio Oct 30 '16 at 12:34
  • $\begingroup$ @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∥ $\endgroup$ – lio Oct 30 '16 at 14:19
  • $\begingroup$ 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. $\endgroup$ – DonAntonio Oct 30 '16 at 15:23
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$$\left\|4x^2-1\right\|^2=\langle 4x^2-1,\,4x^2-1\rangle=4\langle x^2,\,x^2\rangle-8\langle x^2,\,1\rangle+\langle 1,\,1\rangle$$

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.

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  • $\begingroup$ I think part of the point is that $a,b,c$ have to be chosen carefully to make the thing an inner product. $\endgroup$ – Gerry Myerson Oct 30 '16 at 11:53
  • $\begingroup$ @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 $\;a,b,c\;$ . $\endgroup$ – DonAntonio Oct 30 '16 at 12:33
  • $\begingroup$ @GerryMyerson I already edit the question.I misinterpreted the question maybe. can you help me to answer this question? $\endgroup$ – lio Oct 30 '16 at 14:26

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