The range (image, column space) is the set that some function T maps to. constituents some T: V\to W requirements The range is just the space the map maps to:
\begin{equation} range\ T = \{Tv: v \in V\} \end{equation}
additional information range is a subspace of the codomain This result is hopefully not super surprising. zero \begin{equation} T0 = 0 \end{equation} as linear maps take 0 to 0, so 0 is definitely in the range. addition and scalar multiplication inherits from additivity and homogeneity of Linear Maps. Given T v_1 = w_1,\ T v_2=w_2, we have that w_1, w_2 \in range\ T.
\begin{equation} T(v_1 + v_2) = w_1 + w_2 \end{equation}
\begin{equation} T(\lambda v_1) = \lambda w_1 \end{equation}
So closed under addition and scalar multiplication. Having shown the zero and closure, we have that the range is a subspace of the codomain. \blacksquare