Let $T:V \rightarrow W$ be a linear map. $Im(T)$ be the image of $T$. $Im(T) \subset W$

Does the equalitiy of the dimension of $Im(T)$ is and the dimension of $W$ imply that $Im(T) = W$?

Question

Let $T:V \rightarrow W$ be a linear map. $Im(T)$ be the image of $T$. $Im(T) \subset W$

Does the equalitiy of the dimension of $Im(T)$ is and the dimension of $W$ imply that $Im(T) = W$?

compphysgeek · Answer

I guess I can give the answer myself. 

Let W be two-dimensional without loss of generality. Any two-dimensional subspace of W must be W itself.

So the answer must be Yes.