What is the output of transformation by a $3 × 2$ matrix?

Question

I know a $3 × 2$ matrix transforms $2$-dimensional vectors to $3$-dimensional vectors. So the input could be a plane, but what about output? Is it always a plane in a $3$-dimensional space? I know the column space of the matrix is $2$.

Answer 1

It is indeed a plane in a 3D space, but not just any plane: it is a plane that contains the origin $(0,0,0)$, since a linear transformation maps a zero vector to a zero vector.