Whether the image is a subspace of a linear transformation $V$ and $ W $are two real vector spaces.
$T: V \rightarrow W$ is a linear transformation.
What is the image of $T$ and how can I prove that it is a subspace of W?
 A: The image of the linear transform $T: V \to W$ is simply the set of all points in $W$ that are "reached" / affected by $T$.
More formally,
$$
Im(T) = \{ T(v)\ | \ v \in V\} 
$$
For example, consider the linear transform 
$$
T: [0, 1] \to \mathbb R \\
T(x) = x + 1
$$
The image of $T$ will be $[1, 2]$ even though the codomain in $\mathbb R$, since all of $\mathbb R$ is not reached by $T$ acting on $[0, 1]$.
The image of $T$  is clearly a subset of $W$, so all you need to do is to show that it is a subspace. To do this, you need to show that the linear combination of any two elements in $W$ is still an element in $W$.
Let $w_1, w_2 \in Im (T)$ and let $r_1, r_2 \in \mathbb R$. Consider 
$$
r_1w_1 + r_2w_2
$$
Since $w_1, w_2 \in Im(T)$, that means that there exists $v_1, v_2 \in V$ such that $w_1 = T(v_1)$ and $w_2 = T(v_2)$. Hence, rewrite the original expression as
$$
r_1 T(v_1) + r_2 T(v_2) = \\
T(r_1v_1) + T(r_2v_2) = \\
T(r_1v_1 + r_2v_2)
$$
But $T(r_1v_1 + r_2v_2) \in Im(T)$, and hence the linear combination of $w_1$ and $w_2$ is in $Im(T)$. Hence, $Im(T)$ is a subspace of $W$.
