Question about 3blue1brown's vector basis video

Question

I've just watched this video https://www.youtube.com/watch?v=P2LTAUO1TdA, I was wondering, if we have basis \begin{bmatrix}2&-1\\1&1\end{bmatrix} When we take linear combinations of these basis' the columns correspond to both points in the x and y direction, i.e. the first column is the first vector that is 2 in the x direction and 1 in the y, but multiplying by vector \begin{bmatrix}x \\y \end{bmatrix} to give the linear system of equations \begin{bmatrix}2x-y\\x+y\end{bmatrix} the combination of the columns correspond to points in only one direction. Not sure if I've explained this well, any help would be appreciated, thanks.

Answer 1

It might be useful to look at the matrix product in the following way, which some of the comments have basically sketched out:

$$ \begin{bmatrix} 2 & -1 \\ 1 & 1 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ \end{bmatrix} \rightarrow x \begin{bmatrix} 2 \\ 1 \\ \end{bmatrix} +y \begin{bmatrix} -1 \\ 1 \end{bmatrix} $$

This way, you can directly see that you still have a linear combination of the basis, but now the coefficients are $x$ and $y$ which are your unknowns. I hope this helps.