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# Midterm 2: 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 space

- What is the definition of a vector space? Provide some examples other than those you saw on the textbook.
- Is $V = \left\{\begin{bmatrix} x \\ y \end{bmatrix} : x^2 + y^2 \leq 1 \right\}$ a vector space? Why?
- Is $V = \left\{\begin{bmatrix} x \\ y \end{bmatrix} : x + y = 0 \right\}$ a vector space? Why?
- Is $V = \left\{\begin{bmatrix} x \\ y \end{bmatrix} : x + y = 1 \right\}$ a vector space? Why?
- Is $V = \left\{\begin{bmatrix} x \\ y \end{bmatrix} : y > 0 \right\}$ a vector space? Why?

Related resources: `Hefferon Two.I.1`, `LR 15`, `NB 208`

## Subspace

- What is the definition of a subspace? Provide some examples other than those you saw on the textbook.
- Given a subset of a vector space. How do we verify if it is a subspace?
- Is $\emptyset$ a subspace of $\mathbb{R}^2$? Why?
- Is $V = \left\{\begin{bmatrix} x \\ y \end{bmatrix} : x^2 + y^2 \leq 1 \right\}$ a subspace of $\mathbb{R}^2$? Why?
- Is $V = \left\{\begin{bmatrix} x \\ y \end{bmatrix} : x + y = 0 \right\}$ a subspace of $\mathbb{R}^2$? Why?
- Is $V = \left\{\begin{bmatrix} x \\ y \end{bmatrix} : x + y = 1 \right\}$ a subspace of $\mathbb{R}^2$? Why?
- Is $V = \left\{\begin{bmatrix} x \\ y \end{bmatrix} : y > 0 \right\}$ a subspace of $\mathbb{R}^2$? Why?

Related resources: `Hefferon Two.I.2`, `NB 209`

## Linear independence

- What is the definition of a set of vector being linearly independent?
- How to show if a set of vectors $S$ is linearly independent?
- How to show if a set of vectors $S$ is linearly dependent?
- Given a matrix $A$ and its kernel, how to tell if a set of columns of $A$ is linearly independent.

Related resources: `Hefferon Two.II`, `LR 17~19`, `NB 201`

## Basis and dimension

- What is the definition of a basis of a vector space?
- Find a basis of $\mathbb{R}^n$.
- Find a basis of $V = \left\{\begin{bmatrix} x \\ y \\ z \end{bmatrix} : x + y + z = 0 \right\}$.
- What is the definition of the dimension of a vector space?
- Given a matrix $A$, how to find a basis of $\Row(A)$? What is its dimension?
- Given a matrix $A$, how to find a basis of $\ker(A)$? What is its dimension?
- Given a matrix $A$, how to find a basis of $\Col(A)$? What is its dimension?
- What is the definition of $\rank(A)$ and $\nul(A)$?

Related resources: `Hefferon Two.III.1~3`, `LR 20~22`, `NB 202~204, 207`

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