If $T\colon V\to W$ is a linear transformation, and $N$ is the nullspace of $T$, then for every subspace $A$ of $V$ we have $\dim(T(A))+\dim(A\cap N) = \dim (A).$ This is a simple consequence of the Rank-Nullity theorem.
In particular, since the restriction of $T$ to $A$ is one-to-one if and only if $A\cap N = \{\mathbf{0}\}$, it follows that $\dim(T(A))=\dim(A)$ if and only if the restriction of $T$ to $A$ is one-to-one, if and only if $A\cap N=\{\mathbf{0}\}$.
(This is actually a special case of the homomorphism theorems, applied to linear algebra; the image of $A$ is the same as the image of $A+N$, which is isomorphic to $(A+N)/N \cong A/(A\cap N)$.