I have a $T\colon\mathbb{R}^2\to\mathbb{R}^2$ linear operator given by $$T(x,y) = (2x - y, -8x + 4y)$$
How can I tell if the vector $(1, -4)$ is in $R(T)$?
Ok so I set everything into a matrix:
$$\left( \begin{matrix} 2 & -1 & 1\\ -8 & 4 & -4\\ \end{matrix}\right) $$
I row reduced it and found that it was linearly dependent. So I'm going to assume that that means it's is in R(T).
Edit: I'm not sure if I did that right because it isn't set to 0.