$$

Midterm 1: Questions to ponder

Please see Course website for details about the exam.

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 by Jim Hefferon
   • NB Linear algebra notebook by Jephian Lin
   • LR Learning resources
4. Think carefully whether your answer is correct or not.
5. 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
  $S=\{{\mathbf{u}}_1,\dots ,{\mathbf{u}}_d\}$, what is a linear combination of
  $S$?
• Given a set of vectors
  $S=\{{\mathbf{u}}_1,\dots ,{\mathbf{u}}_d\}$, what is the definition of
  $\mathrm{span}(S)$ under the set-builder notation?
• Let
  $S=\left\{\left[\begin{array}{c}1\\ 2\end{array}\right]\right\}$. How does
  $\mathrm{span}(S)$ 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
  $\mathbf{b}$, how to show
  $\mathbf{b}$ is in
  $\mathrm{span}(S)$?
• Given a set of vector
  $S$ and a vector
  $\mathbf{b}$, how to show
  $\mathbf{b}$ is not in
  $\mathrm{span}(S)$?
• Given a set of vector
  $S$, a vector
  $\mathbf{p}$, and a vector
  $\mathbf{b}$, how to show
  $\mathbf{b}$ is in
  $\mathbf{p}+\mathrm{span}(S)$?
• Given a set of vector
  $S$, a vector
  $\mathbf{p}$, and a vector
  $\mathbf{b}$, how to show
  $\mathbf{b}$ is not in
  $\mathbf{p}+\mathrm{span}(S)$?

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
  $\mathrm{Row}(A)$ of a matrix
  $A$?
• How to determine if a vector is in
  $\mathrm{Row}(A)$?
• What is the relation between
  $\ker (A)$ and
  $\mathrm{Row}(A)$?
• How to show that
  $\mathrm{span}(\{\mathbf{x},\mathbf{y},\mathbf{z}\})=\mathrm{span}(\{\mathbf{x},\mathbf{y},\mathbf{y}+\mathbf{z}\})$?

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

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