I've a $2\times2$ non-invertible matrix . It maps every point into a line.
How can I prove that for every single vector $v$ in the codomain i can find a line $l$ in the domain such as the matrix maps every vector in the line $l$ into the vector $v$ ?
Let $v$ be an image of the map (as pointed out by amd, we need it to be image rather than the codomain).
Consider the system
$$Ax=v.\tag{1}$$
We already know that the system is consistent.
Hence there exists an $x_0$ such that $Ax_0=v$, also remember that $A$ is a rank $1$ matrix, think of what we can say about the kernel and the general equation to $(1)$.