What is the range of a linear transformation? And when is that transformation onto?
For a linear transformation $T \in \mathcal{L} (V, W)$, we refer to $\range(T)$ as the range space (or image space) of $T$. The language is justified by the following proposition.
For any linear map we have $T(0) = 0$, since scalar multipliciation commutes with the application of $T$. (We can express the origin $0 \in W$ as the scalar $0$ times the origin.) Hence, the origin in $V$ is mapped to the origin in $W$ and so $0 \in W$.
To see that $W$ is closed under vector
addition, let $w_1, w_2 \in \range T \subset W$
Then there exists $v_1, v_2 \in V$ so that
\[
Tv_1 = w_1 \text{ and } Tv_2 = w_2
\] \[
w_1 + w_2 = Tv_1 + Tv_2 = T(v_1 + v_2)
\]
Similarly, to see that $W$ is closed under
scalar multiplication, let $w \in \range T$ and
$\lambda \in \F $.
Here $\F $ denotes the field over $W$ and $V$.
Then there exists $v \in V$ satisfying $w =
Tv$.
We claim
\[
\lambda w = \lambda T(v) = T(\lambda v)
\]