show that a transformation is linear

Question

show that the given transformation is linear: $$T((1,0)) = (1,2,0)$$ $$T((0,1)) = (1,0,-1)$$ I know how to generally show a transformation is linear. But i am not sure how to do it if the transformation is given like that.

Answer 1

I'm assuming you have to find the linear transformation which gives these results.

So we have $ T(e_1) = (1,2,0)$ and $T(e_2) = (1,0,-1)$. Since the inputs form a basis for $\mathbb{R}^2$ then the transform is

$$T = \begin{bmatrix} 1 & 1 \\ 2 & 0 \\ 0 & -1 \end{bmatrix} \>.$$

Can you see why this is the linear transform?