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

  • $\begingroup$ @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

$$\left\|4x^2-1\right\|^2=\langle 4x^2-1,\,4x^2-1\rangle=4\langle x^2,\,x^2\rangle-8\langle x^2,\,1\rangle+\langle 1,\,1\rangle$$

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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.