I have only started linear algebra, I was watching the lectures of Gilbert Strang, he gave us two linear equation
$2x-y=0$
$x-2y=3$ .
He wrote this in the way of coefficient matrices and it became
$\begin{bmatrix} 2 & -1 \\ 1 & -2 \end{bmatrix} $ $\begin{bmatrix} x \\ y \end{bmatrix} $=$\begin{bmatrix} 0\\ 3 \end{bmatrix}$.
Now he wrote this as $\begin{bmatrix}2 \\ 1 \end{bmatrix}$$x$ +$\begin{bmatrix} -1\\ -2 \end{bmatrix}$$y$ =$\begin{bmatrix} 0 \\ 3 \end{bmatrix}$.
Next he made the matrix \begin{bmatrix} 2\\ 1 \end{bmatrix} into a vector with components $2$ and $1$. Now I do not get how can you do that aren't these just coefficients matrices which are basically scalars multiplied by components of vectors which are $x$ and $y$. Can someone tell why has it been done like this??
$$\begin{bmatrix} x \\ y \end{bmatrix} =x\begin{bmatrix} 1\\ 0 \end{bmatrix} +y\begin{bmatrix} 0\\ 1 \end{bmatrix} $$
So $$ \begin{bmatrix} 2 & -1 \\ 1 & -2 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} =x \begin{bmatrix} 2 & -1 \\ 1 & -2 \end{bmatrix} \begin{bmatrix} 1\\ 0 \end{bmatrix} +y\begin{bmatrix} 2 & -1 \\ 1 & -2 \end{bmatrix}\begin{bmatrix} 0\\ 1 \end{bmatrix} $$
$$= x \begin{bmatrix} 2 \\ 1 \end{bmatrix} +y\begin{bmatrix} -1 \\ -2 \end{bmatrix} $$
Don't think of $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ as two of the four components in the matrix, think of it as the vector you get when you use the matrix to transform the unit vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$