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# Midterm 1: 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

- What is the definition of $\mathbb{R}^n$ under the set-builder notation?
- How to calculate the length of a given vector $\mathbb{R}^n$?
- How to calculate the distance between two points in $\mathbb{R}^n$?
- How to calculate the angle between two vectors in $\mathbb{R}^n$ using the law of cosine?
- How to calculate the angle between two vectors in $\mathbb{R}^n$ using the inner product?
- Do the two definitions of the angle always give the same answer?

Related resources: `Hefferon One.II`, `NB 101`

## Subspace

- Given a set of vectors $S = \{\bu_1, \ldots, \bu_d\}$, what is a linear combination of $S$?
- Given a set of vectors $S = \{\bu_1, \ldots, \bu_d\}$, what is the definition of $\vspan(S)$ under the set-builder notation?
- Let $S = \left\{\begin{bmatrix} 1 \\ 2 \end{bmatrix}\right\}$. How does $\vspan(S)$ look like on the plane of $\mathbb{R}^2$? Try the same question with a different set $S$.
- Given a set of vector $S$ and a vector $\bb$, how to show $\bb$ is in $\vspan(S)$?
- Given a set of vector $S$ and a vector $\bb$, how to show $\bb$ is not in $\vspan(S)$?
- Given a set of vector $S$, a vector $\bp$, and a vector $\bb$, how to show $\bb$ is in $\bp + \vspan(S)$?
- Given a set of vector $S$, a vector $\bp$, and a vector $\bb$, how to show $\bb$ is not in $\bp + \vspan(S)$?

Related resources: `NB 102, 106`, `LR 1~5`,

## System of linear equations

- How to solve a system of linear equations completely?
- What is $\ker(A)$ of a matrix $A$?
- How to determine if a vector is in $\ker(A)$?
- What is $\Row(A)$ of a matrix $A$?
- How to determine if a vector is in $\Row(A)$?
- What is the relation between $\ker(A)$ and $\Row(A)$?
- How to show that $\vspan(\{\bx,\by,\bz\}) = \vspan(\{\bx,\by,\by+\bz\})$?

Related resources: `LR 6~10`, `NB 102~104, 107~111`

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