MAT223 Linear Algebra Exam Review questions answered in videos

Dear students from MAT223:

I’ve received some requests to explain the following questions and I made a few short videos to explain how to solve them.

Happy learning!

  1. If $T$ and $S$ are linear transformations $ \mathbb R^3 \rightarrow \mathbb R^3 $, and $\displaystyle \vec{u}, \vec{v}, \vec{w}$ are three distinct vectors in $\mathbb R^3$ so that

 T(\vec{u})= S(\vec{u}), T(\vec{v})= S(\vec{v}), T(\vec{w})= S(\vec{w}),

then $\displaystyle T(\vec{x}) =S(\vec{x}) $ for all $ \displaystyle \vec{x} \in \mathbb R^3.$

2. If A is 3 \times 3 matrix and dim(null(A-I))=3, then A is diagonalizable.

3. Consider

     \begin{equation*}A=\left[ \begin{array}{rrr}2 & 4 & -1  \\0 & 7 & 2  \\0 & -2 & 3\end{array} \right]\end{equation*}

Decide whether A is diagonalizable. Justify your answer.

4. What is the matrix of a linear transformation

 T : \mathbb R^4 \rightarrow \mathbb R^4


    \begin{equation*} T(\vec{e_1})  =\vec{e_2} = 3\vec{e_3}; \\ T(\vec{e_2}) =2 \vec{e_1} - 3\vec{e_2} + \vec{e_3};\\ T(\vec{e_3})  =3 \vec{e_3};\\ T(\vec{e_4}) =-\vec{e_1} - 4\vec{e_4} \end{equation*}

Compute the rank of this matrix.

2019 Winter MAT102: Intro to Math Proof

The main topics that we cover in this course are:

  1. Numbers, and Inequalities.
  2. Sets, functions and fields.
  3. Informal logic.
  4. Mathematical Induction.
  5. Bijections and cardinality.
  6. Integers and divisibility.
  7. Relations.

The course note is available: MAT102

The slides for weekly lectures can be found here:

  1. MAT102_Week1-2
  2. MAT102_Week2-3
  3. MAT102_Week 4-5
  4. MAT102_Week 6
  5. MAT102_Week 7
  6. MAT102_Week 8
  7. MAT102_Week 9
  8. MAT102_Week10
  9. MAT102_Week 11-12

Below are a few short video clips I made to explain certain questions upon students’ request.

  1. 2.5.29 Part a) Prove or disprove if two functions f and g are bounded, then their sum f+g is bounded.
  2. 2.5.29 Part b). Prove of disprove if two functions f and g are bounded, then f^2-g^2 is bounded.
  3. 2.5.50. Show that in any field F, the equation x^2 = 1 can have at most two solutions.
    Can you think of a field in which the equation x^2 = 1 has exactly one solution?