Suggest edit — range

Title

Name

Note

---
title: "range"
type: concept
related: [Zero, Function, Linear Map, Scalar Multiplication, Subspace]
source: https://www.jemoka.com/posts/kbhrange/
confidence: high
status: active
---

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. \(\blacksquare\)