8

log8.pdf

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Answer the following problem:

Let \(T : R^2[x] → R^2[x]\) be a linear operator satisfying \([T]_B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{pmatrix}\), where

\(B = \{1, 1 + x, x^2\}\). Answer the following section without calculating \(T\).

1. Determine if \(T\) is invertible.

   No, Since the corresponding matrix is not row reduced echelon form of identity matrix.

2. Find a basis and dimension to the image and the kernel.

   We have the matrix of linear map, then we can choose the column which has the pivot, that is the image

   Then the remaining column is the null space

   The basis of image is \(\{1+x^2,1+x+x^2\}\) and the dimension of image is \(2\)​

   The basis of kernel is \(\{x^2\}\) and the dimension of kernel is \(1\)

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Answer the following questions:

1. Rate the difficulty of the given problem.
   (1 - very easy, 9 - very hard)

   1 2 3 4 5 6 7 8 9

   What helped you solve the problem?

   The lecture note tells me how to choose column to get the null space and image

2. Suppose the problem is given in the exam, and is worth 20 points. How many points do you think you gained for your solution? Please explain.

   15

   I think i need write more notation to get the result instead of using the trick. However, this is taught in the lecture, thus i think it's ok

3. Do you agree with the following statements?
   (1 - don’t agree at all, 9 - strongly agree)

   1. The purpose of the problem is to evaluate algorithmic skills, such as arithmetic calculations, technical skills, solving methods, etc.
      1 2 3 4 5 6 7 8 9
   2. The purpose of the problem is to evaluate cognitive skills, such as intuitive understanding, train of thought, strategy, etc.
      1 2 3 4 5 6 7 8 9
   3. The purpose of the problem is to evaluate formal skills, such as math-ematical logic, proof structure, proper writing, etc.
      1 2 3 4 5 6 7 8 9

4. 1. Rate the difficulty of your studies in GTIIT so far. (1 - very easy, 9 - very hard)
      1 2 3 4 5 6 7 8 9
      Give an example of something that you find hard:

      Isomorphism......🙃😅

      Dual space......😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃😅🙃 6. Rate your interest in your studies in GTIIT so far. (1 - not interesting at all, 9 - very interesting)
      1 2 3 4 5 6 7 8 9
      Give an example of something that you find interesting:

      Not specific, but i find i construct the base of maths gradually. From the basic logic, to the hard part. Every theorem, corollary, proposition.......that i used i know i all can prove it. This is a amazing experience.