The dimension of the operator if the domain has dimension 2

Question

Suppose $A$ is a linear operator s.t. $A\colon X\rightarrow Y$. If $\dim(X)=2$, what is $\dim(A(X))$?

Answer 1

Necessarily, $\mathrm{dim}(A(X)) \leq 2$. If you choose a basis $\{e_{1}, e_{2}\}$ for $X$, then by linearity of $A$ we have that $A(X) = \mathrm{span}\{Ae_{1}, Ae_{2}\}$, which would have dimension $2$ precisely when $Ae_{1}$ and $Ae_{2}$ are independent.