$$

Midterm 2: 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 space

• What is the definition of a vector space? Provide some examples other than those you saw on the textbook.
• Is
  $V=\{\left[\begin{array}{c}x\\ y\end{array}\right]:x^2+y^2\le 1\}$ a vector space? Why?
• Is
  $V=\{\left[\begin{array}{c}x\\ y\end{array}\right]:x+y=0\}$ a vector space? Why?
• Is
  $V=\{\left[\begin{array}{c}x\\ y\end{array}\right]:x+y=1\}$ a vector space? Why?
• Is
  $V=\{\left[\begin{array}{c}x\\ y\end{array}\right]:y>0\}$ 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
  $\mathrm{\varnothing }$ a subspace of
  ${\mathbb{R}}^2$? Why?
• Is
  $V=\{\left[\begin{array}{c}x\\ y\end{array}\right]:x^2+y^2\le 1\}$ a subspace of
  ${\mathbb{R}}^2$? Why?
• Is
  $V=\{\left[\begin{array}{c}x\\ y\end{array}\right]:x+y=0\}$ a subspace of
  ${\mathbb{R}}^2$? Why?
• Is
  $V=\{\left[\begin{array}{c}x\\ y\end{array}\right]:x+y=1\}$ a subspace of
  ${\mathbb{R}}^2$? Why?
• Is
  $V=\{\left[\begin{array}{c}x\\ y\end{array}\right]:y>0\}$ 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
  $V=\{\left[\begin{array}{c}x\\ y\\ z\end{array}\right]:x+y+z=0\}$.
• What is the definition of the dimension of a vector space?
• Given a matrix
  $A$, how to find a basis of
  $\mathrm{Row}(A)$? 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
  $\mathrm{Col}(A)$? What is its dimension?
• What is the definition of
  $\mathrm{rank}(A)$ and
  $\mathrm{null}(A)$?

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

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