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

with

    \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.

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