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Problem

  • Let $T : \mathbb{P_2\to R}$ be the function defined by $T(p(x))= p(1)$. Verify that $T$ is in fact a linear transformation. (There are two things you need to verify for this.)
  • Let $T$ be the transformation defined above. Find a basis for the kernel of $T$ that is orthogonal with respect to the inner product.$$ \langle f(x),g(x)\rangle =\int_0^1f(x)g(x)\mathrm{dx}$$ (That is, find a basis for the kernel of $T$ such that all pairs of distinct basis vectors are orthogonal with respect to this inner product.)

What is the question asking in the second part? How to find a basis for the kernel of $T$ that is orthogonal with respect to an inner product?

Rohit Singh
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Hello
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  • Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community May 10 '22 at 12:31
  • Don't post images, use mathjax instead to write formulas. There's mathjax tutorial on this site. 2) Do you know about the Gram-Schmidt process?
    • – lisyarus May 10 '22 at 12:32
  • I would seem like the image explains what is being asked. Please explain what you don't understand of that explanation. – Marc van Leeuwen May 10 '22 at 12:37
  • This answer: https://math.stackexchange.com/a/4434547/1027216 should be useful. You should read it and then you can show your work here. – A. P. May 10 '22 at 13:01
  • The kernel of $T$ contains all $p\in\mathbb P_2$ with $T(p)=p(1) =0$. I guess $\mathbb P_2$ means "space of polynomials with degree less or equal to two"? Then $\dim\mathbb P_2=3$ and $\dim \text{im} T=\dim \mathbb R=1$. Hence $\dim \text{ker} T=2$. Find an one-degree polynomial with $p(1) =0$. – Jochen Jun 03 '22 at 15:29