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# Midterm 2: Questions to ponder

Please see [Course website](https://www.math.nsysu.edu.tw/~chlin/2024SMath104A/2024SMath104A.html) for details about the exam.

:::success
:bulb: You need to provide the reasons for your answers.
:::

## How to prepare the exam?

1. Make a **concrete setting** for the problems whenever possible. For example, what is the length of $(1,2,3)$?
2. **Write down** your answer and reasons on a paper.
3. If you do not know how to answer it, **look it up** from the following resources, or ask ChatGPT.
- `Hefferon` [_Linear Algebra_](http://joshua.smcvt.edu/linearalgebra/book.pdf) by Jim Hefferon
- `NB` [_Linear algebra notebook_](https://jephianlin.github.io/LA-notebook/index-en.html) by Jephian Lin
- `LR` [Learning resources](https://hackmd.io/@jephianlin/2024SMath104A-resources)
5. **Think carefully** whether your answer is correct or not.
6. Repeat the above steps with **different settings**.

## Matrix representation

- Describe the meaning of the matrix representation $[f]_\alpha^\beta$.
- Explain why $A = [f_A]_{\mathcal{E}_n}^{\mathcal{E}_n}$, where $\mathcal{E}_n$ is the standard basis of $\mathbb{R}^n$.
- Explain why $[\idmap]_\beta^{\mathcal{E}_n}$ is the matrix whose columns are vectors in $\beta$.
- Given $[\idmap]_\beta^{\mathcal{E}_n}$, how to find $[\idmap]_{\mathcal{E}_n}^\beta$?
- Given a matrix $A$ and a basis $\beta$, how to find $[f_A]_\beta^\beta$?

Related resources: [LR of Math103A](https://hackmd.io/@jephianlin/2023FMath103A-resources) 31~35

## Diagonalization

- Explain why $[f_A]_\beta^\beta$ is a diagonal matrix implies $\beta$ is composed of vectors $\bv$ with $A\bv = \lambda\bv$ for some scalar $\lambda$.
- Describe the definition of the eigenvalue and the eigenvector.
- How to calculate the eigenvalues of a matrix?
- How to calculate the eigenvectors of a matrix?
- Describe the definition of the characteristic polynomial.
- Explain why the eigenvalues are the roots of the characteristic polynomial.
- Describe the definition of the algebraic multiplicity and the geometric multiplicity.
- How to determine if a matrix is diagonalizable or not?
- How to find the appropriate axes for a quadratic curve?

Related resources: `Hefferon Five.II`, `LR 19~23`, `NB 501, 502, 506, 508~510, `

## Characteristic polynomial

- How to calculate the coefficients of the characteristic polynomial of a matrix?
- What is the relation between the trace $\tr(A)$ and the eigenvalues of $A$?
- What is the relation between the determinant $\det(A)$ and the eigenvalues of $A$?

Related resources: `Hefferon Five.II`, `LR 24~27`, `NB 506, 507`

*This note can be found at Course website > Learning resources.*