Doubt about Linear Transformations

Question

A linear transformation $T:V->W$ is surjective if $Range(T)=W$. Now let A and B be two vector spaces such that A is a subspace of B and $dim(A)=dim(B)$ ; we can say that $A=B$. So from this can we say that T is surjective only when $dim(Range(T))=dim(W)$ because Range(T) is a subspace of W ?

Answer 1

Yes, if you have a finite-dimensional vector space $\textsf V$ and $\textsf W \subseteq \textsf V$ a subspace of it, then : $$\begin{matrix} \dim (\textsf W) < \dim (\textsf V) & \Leftrightarrow & \textsf W \subsetneq \textsf V \\ \dim (\textsf W) = \dim (\textsf V) & \Leftrightarrow & \textsf W = \textsf V \end{matrix}$$