Section 1 The Dimension Formula
Q1.4
Prove that every $m \times n$ matrix $A$ of rank 1 has the form $A=X Y^{\mathrm{t}}$, where $X, Y$ are $m$ - and $n$-dimensional column vectors. How uniquely determined are these vectors?
We have:
\[A=X Y^{t}=\left[\begin{array}{c} \mid \\ X \\ \mid \end{array}\right]\left[\begin{array}{lll} y_{1} \cdots & y_{n} \end{array}\right]=\left[\begin{array}{cccc} \mid & \mid & & \mid \\ y_{1} X & y_{2} X & \cdots & y_{n} X \\ \mid & \mid & & \mid \end{array}\right]\]The column space formed by $X Y^t$ always has dimension 1, because all columns are scalar multiples of $X,$ therefore dependent on $X$ and $A$ has rank 1. Vice versa.
These vectors are uniquely determined up to scalar multiplication.
For example, if we have $X, Y$ such that $A=X Y^t$,
then if we let $X^{\prime}=c X, Y^{\prime}=\frac{1}{c} Y$
$\Rightarrow A=X^{\prime} {Y^{\prime}}^t=c X \cdot \frac{1}{c} Y^{t}=X Y^{t}$,
still holds for $X^{\prime}$ and $Y^{\prime}.$
Section 2 The Matrix of a Linear Transformation
Q2.1
Let $A$ and $B$ be $2 \times 2$ matrices. Determine the matrix of the operator $T: M \leadsto A M B$ on the space $F^{2 \times 2}$ of $2 \times 2$ matrices, with respect to the basis $\left(e_{11}, e_{12}, e_{21}, e_{22}\right)$ of $F^{2 \times 2}$.
Let \(A=\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right), B=\left(\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right)\)
By direct calculation,
\[T\left(e_{11}\right)=A e_{11} B=\left(\begin{array}{ll}a_{11} b_{11} & a_{11} b_{12} \\ a_{21} b_{11} & a_{21} b_{12}\end{array}\right)\] \[T\left(e_{12}\right)=\left(\begin{array}{ll}a_{11} b_{21} & a_{11} b_{22} \\ a_{21} b_{21} & a_{21} b_{22}\end{array}\right), T\left(e_{21}\right)=\left(\begin{array}{ll}a_{12} b_{11} & a_{12} b_{12} \\ a_{22} b_{11} & a_{22} b_{12}\end{array}\right)\] \[T\left(e_{22}\right)=\left(\begin{array}{ll}a_{12} b_{21} & a_{12} b_{22} \\ a_{22} b_{21} & a_{22} b_{22}\end{array}\right)\]So \(T\left(e_{11}\right)=\left(e_{11}, e_{12}, e_{21}, e_{22}\right)\left(\begin{array}{l}a_{11} b_{11} \\ a_{11} b_{12} \\ a_{21} b_{11} \\ a_{21} b_{12}\end{array}\right)=\mathbf{C} A_{1}\) Where $\mathbf{C}$ is the basis, $A_{1}$ is the first column of the matrix of the operator.
Similarly for $T\left(e_{12}\right),T\left(e_{21}\right),T\left(e_{22}\right)$. so the mathix of operator, $A$, is obtained by assembling $A_1,A_2,A_3,A_4$ into the columns of $A$ by the formula: $T(\mathbf{C})=\mathbf{C}A$
\[A=\left(\begin{array}{llll}a_{11} b_{11} & a_{11} b_{21} & a_{12} b_{11} & a_{12} b_{21} \\ a_{11} b_{12} & a_{11} b_{22} & a_{12} b_{12} & a_{12} b_{22} \\ a_{21} b_{11} & a_{21} b_{21} & a_{22} b_{11} & a_{22} b_{21} \\ a_{21} b_{12} & a_{21} b_{22} & a_{22} b_{12} & a_{22} b_{22}\end{array}\right)\]Q2.2
Let $A$ be an $n \times n$ matrix. and let $V$ denote the space of $n$-dimensional row vectors. What is the matrix of the linear operator “right multiplication by $A$ “ with respect to the standard basis of $V$ ?
The standard basis is $\mathbf{E}=\left(e_{1}, \cdots, e_{n}\right)$,
Let $A=\left(a_{i j}\right)$,
$T\left(e_{j}\right)=e_{j} A=A_{j}$, where $A_{j}$ stands for the j-th row of $A$.
\[\Rightarrow T\left(e_{j}\right)=A_{j}=a_{j 1} e_{1}+\cdots+a_{j n} e_{n}=\left(e_{1}, \cdots, e_{n}\right)\left( \begin{array}{c} a_{j 1} \\ \vdots \\ a_{j n} \end{array}\right)=\mathbf{E} M_{j}\]$M_{j}$ is the $j$ th column of $M$, the matrix of the linear operator. Assemble all $M_{j}$ into columns, we have,
\[M=\left(\begin{array}{ccccc} a_{11} & \cdots & a_{j 1} & \cdots & a_{n 1} \\ \vdots & & \vdots & & \vdots \\ a_{1 n}& \cdots & a_{j n} & \cdots & a_{n n} \end{array}\right)\]So $M=A^t$, the transpose of $A$ is the matrix of the linear operator “right multiplication by $A$”.
Q2.3
Find all real $2 \times 2$ matrices that carry the line $y=x$ to the line $y=3 x$.
Using the standard basis, any vector on $y=x$ is of the form $k(1,1)^{t}$, under a $2 \times 2$ matrix multiplication,
\[k\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)\left(\begin{array}{l}1 \\ 1\end{array}\right)=k\left(\begin{array}{l}a+b \\ c+d\end{array}\right)\]To ensure $k(a+b, c+d)^{t}$ is on $y=3x$, simply let $a+b=3(c+d)$. Any $2 \times 2$ real matrices satisfying this rule will work.