$$

Final: 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 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
  $\{\left[\begin{array}{c}x\\ y\\ z\end{array}\right]:x+y+z=0\}\}$.
• Verify that
  $\mathbf{v}\mapsto [\mathbf{v}{]}_{\beta }$ 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
  $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$, find a matrix
  $A$ such that
  $f(\mathbf{x})=A\mathbf{x}$ for all
  $\mathbf{x}\in {\mathbb{R}}^n$.
• Given the matrix representation
  $[f{]}_{\alpha }^{\beta }$, find
  $f(\mathbf{x})$.
• Given a linear function
  $f:V\to W$ 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
  $[\mathrm{id}{]}_{\alpha }^{\beta }$.
• Given two bases
  $\alpha ,\beta$ of a vector space
  $V$, find the change of basis matrix
  $[\mathrm{id}{]}_{\alpha }^{\beta }$.
• Given two bases
  $\alpha ,{\alpha }^{\prime }$ of
  $V$ and two bases of
  $\beta ,{\beta }^{\prime }$ of
  $W$, use the matrix representation
  $[f{]}_{\alpha }^{\beta }$ to find
  $[f{]}_{{\alpha }^{\prime }}^{{\beta }^{\prime }}$.

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

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