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# Final: Questions to ponder

Please see [Course website](https://www.math.nsysu.edu.tw/~chlin/2023FMath103A/2023FMath103A.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/2023FMath103A-resources)
5. **Think carefully** whether your answer is correct or not.
6. Repeat the above steps with **different settings**.

## Vector representation

- How to convert a vector into its vector representation?
- How to convert a vector representation back to its original vector?

Related resources: `Hefferon Two.III.1`, `NB 305, 306`

## Linear function

- What is a function?
- Is a function injective?
- Is a function surjective?
- Is a function linear?
- Find an isomorphism between $\mathbb{R}^2$ and $\left\{\begin{bmatrix} x \\ y \\ z \end{bmatrix}: x + y + z = 0 \}\right\}$.
- Verify that $\bv\mapsto[\bv]_\beta$ is an isomorphism.
- Find a basis of the kernel of a linear function.
- Find a basis of the range of a linear function.

Related resources: `Hefferon Three.I~II`, `LR 25, 26, 28, 29, 36`, `NB 301, 302, 303, 304, 308`

## Matrix representation

- Given a linear function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$, find a matrix $A$ such that $f(\bx) = A\bx$ for all $\bx\in\mathbb{R}^n$.
- Given the matrix representation $[f]_\alpha^\beta$, find $f(\bx)$.
- Given a linear function $f:V \rightarrow W$ and bases $\alpha,\beta$ of $V,W$, respectively, find the matrix representation $[f]_\alpha^\beta$.

Related resources: `Hefferon Three.III`, `LR 31, 33, 34`, `NB 309, 310`

## Change of basis

- Given two bases $\alpha,\beta$ of $\mathbb{R}^n$, find the change of basis matrix $[\idmap]_\alpha^\beta$.
- Given two bases $\alpha,\beta$ of a vector space $V$, find the change of basis matrix $[\idmap]_\alpha^\beta$.
- Given two bases $\alpha,\alpha'$ of $V$ and two bases of $\beta,\beta'$ of $W$, use the matrix representation $[f]_\alpha^\beta$ to find $[f]_{\alpha'}^{\beta'}$.

Related resources: `Hefferon Three.V`, `LR 35`, `NB 307`

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