Why do we represent each column by x and y values in column picture of a system of linear equations

Question

I started watching Gilbert Strang's Linear Algebra lectures. In the first lecture, he has the following system of linear equations

$$\begin{cases} 2x - y = 0 \\ -x + 2y = 3 \end{cases} $$

When he talks about column picture of the equations, he writes the equations as the following linear combination of columns

$X\begin{bmatrix} 2 \\ -1\end{bmatrix}$ + $Y\begin{bmatrix} -1 \\ 2\end{bmatrix}$ = $\begin{bmatrix} 0 \\ 3\end{bmatrix}$

And then, he shows how it looks geometrically as follows

One thing that i don't understand from the above representation is, how can we consider two coefficients of X (2 and -1) and plot them as if they are X and Y coordinates of a vector?

Answer 1

Take the following substitutions: $u=\begin{bmatrix} 2 \\ -1\end{bmatrix}$, $v=\begin{bmatrix} -1 \\ 2\end{bmatrix}$ and $w=\begin{bmatrix} 0 \\ 3\end{bmatrix}$. Then, Gilbert Strang says that we have to add some amount (say $x=a$) of the vector $u$ to some amount (say $y=b$) of the vector $v$ to get the vector $w=au+bv$. Next, he finds the right amounts to make the equality true and simply draws them. I don't think there is something perplexing :)

Answer 2

We didn't represent X,Y tables of data in X,Y plots before 1600.

It was a sort of discovery, and you can test it, predict, and so on.

It is useful.

In the same way, you can look at those coefficients as being vectors, and it's very insightful. Because from their shape and values, you can derive properties of solutions etc.

It is exactly the same with matrices.