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log7.pdf

Definition:

Given a matrix \(A \in F^{n \times n}\), a polynomial \(p(A)\) in the matrix \(A\) is \(p(A) = a_m A^m + a_{m-1} A^{m-1} + \dots + a_1 A + a_0 I\)

where \(a_i \in F\) are scalar coefficients and \(I \in F^{n \times n}\) is the identity matrix.

Answer the following problem:

Prove that for every matrix \(A \in F^{n \times n}\) there exists a non-zero polynomial (e.g. there exists a non-zero coefficient \(a_i\)) such that \(p(A) = 0\).

We need to prove \(p(A)=a_{m}A^{m}+a_{m-1}A^{m-1}+\dots+a_1A+a_0I=0\) where \(\exists a_i\neq 0\)

Which means \(\{I,A,A^2,\ldots,A^{m}\}\) is linearly dependent

Since \(\dim A=n^2\), we also know \(p(A)\in F^{n\times n}\) which has the same dimension

Then if \(m>n^2\), the list is definitely linearly dependent, then there must exists non-zero coefficient which can let the \(p(A)=0\)

Answer the following questions:

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

   \(1\ 2\ 3\ 4\ 5\ 6\ 7\boxed 8\ 9\)

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

   5

   I have no clearly idea of this question, i think my proof is poor proof, i have not enough confident to solve it. Even i think it doesn't have relation to my previous class contents.

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

   (a) The purpose of the problem is to evaluate algorithmic skills, such as arithmetic calculations, technical skills, solving methods, etc.

   \(1\ 2\ \boxed{3}\ 4\ 5\ 6\ 7\ 8\ 9\)

   (b) The purpose of the problem is to evaluate cognitive skills, such as intuitive understanding, train of thought, strategy, etc.

   \(1\ 2\ 3\ 4\ 5\ 6\ 7\ \boxed{8}\ 9\)

   (c) The purpose of the problem is to evaluate formal skills, such as mathematical logic, proof structure, proper writing, etc.

   \(1\ 2\ 3\ 4\ 5\ 6\ 7\ \boxed{8}\ 9\)

4. Complete the following sentence:

   “Learning mathematics is important because ... ”

   Maths provides us a useful language of science, engineer, economics, and other fields. Thus it serves as a base for advanced studies in STEM fields, enabling technological innovation and scientific discovery. Moreover, i think maths can train our brain, to make it more flexible and abstract. Also the process of leanring maths will let us learn how to conquer the difficulty and deal with the upset emotion

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