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 \(Im(T) = W\)?

Answer 1

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.