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# Final exam: 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.
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## 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 $[f]_\alpha^\beta$.
- For any $m\times n$ matrix, explain why $A = [f_A]_{\mathcal{E}_n}^{\mathcal{E}_m}$, where $\mathcal{E}_k$ is the standard basis of $\mathbb{R}^k$.
- 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 an $m\times n$ matrix $A$, a basis $\alpha$ of $\mathbb{R}^n$, and a basis $\beta$ of $\mathbb{R}^m$, how to find $[f_A]_\alpha^\beta$?
Related resources: [LR of Math103A](https://hackmd.io/@jephianlin/2023FMath103A-resources) 31~35
## Projection
- Given a subspace $V = \vspan\{\bu\}$ and a vector $\bb$, find the projection of $\bb$ onto $V$.
- Given a subspace $V = \vspan\{\bu_1, \ldots, \bu_d\}$ and a vector $\bb$, find the projection of $\bb$ onto $V$.
- Given a matrix $A$ and a vector $\bb$, find $\bx$ that minimizes $\|A\bx - \bb\|^2$.
- Given $\bx = (x_1, \ldots, x_N)\trans$ and $\by = (y_1, \ldots, y_N)\trans$, find $c_0$ and $c_1$ such that $f(x) = c_0 + c_1x$ minimizes $\sum_{i=1}^N (f(x_i) - y_i)^2$.
Related resources: `Hefferon Three.VI`, `LR 29~33`, `NB 105`
## Spectral theorm
- Define an orthogonal basis and an orthonormal basis.
- Define an orthogonal matrix.
- Given a matrix $A$, how to find an orthonormal basis of $\ker(A)$?
- Describe the spectral theorem.
- Given a real symmetric matrix $A$, find an orthonormal basis composed of eigenvectors that diagonalizes $A$.
- Given a real symmetric matrix $A$, find the spectral decomposition of $A$.
- Describe the meaning of the spectral decomposition.
Related resources: `LR 34~36`, `NB 603, 604`
## Rayleigh quotient
- Define Rayleigh quotient.
- Describe the Rayleigh quotient theorem.
- Given a diagonal matrix $A$, find the maximum Rayleigh quotient value and the vector that achieves it.
- Given a real symmetric matrix $A$, find the maximum Rayleigh quotient value and the vector that achieves it.
Related resources: `LR 37~38`, `NB 609`
## Singular value decomposition
- Describe the singular value decomposition.
- Describe the meaning of the singular value decomposition.
- Explain why $AA\trans$ and $A\trans A$ are symmetric.
- Explain why $AB$ and $BA$ have the same nonzero eigenvalues.
- What is the relations between the eigenvalues of $AA\trans$, or $A\trans A$, and the singular value of $A$?
Related resources: `LR 39~41`, `NB 314, 605`
*This note can be found at Course website > Learning resources.*