We just started the topic of linear transformations and I have this hw question that I just don't understand.
Does there exist a non-trivial linear transformation, represented by some 2x2 matrix, which maps the entire Cartesian plane to the line $L = \{(x,y)\mid x = y\}$
I read up on some of the other linear transformation questions on this website but they were pretty specific questions. Could someone give me the general idea on how to solve a question asking for the existence of a linear transformation and how to prove it? Also an explanation on how to approach this problem would be much appreciated.
You can always define a linear transformation by defining what it does to a basis. In your case $(1,0)$ and $(0,1)$ are a basis for $\mathbb{R}^2$. Just define a linear transformation by deciding where on the line $y=x$ they should go.