Houjun Liu


The range (image, column space) is the set that some function \(T\) maps to.


some \(T: V\to W\)


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.


\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. \(\blacksquare\)