The range of a linear transformation is the set of all possible outputs produced when the transformation is applied to every vector in its domain. If a transformation maps vectors from a vector space (V) into a vector space (W), its range tells us which vectors in (W) can actually be reached. Understanding this concept is essential for studying matrix transformations, systems of linear equations, dimension, rank, and whether a transformation is onto.
Introduction
A linear transformation provides a rule that takes an input vector and produces an output vector while preserving vector addition and scalar multiplication. That said, the transformation may not produce every vector in its stated codomain. The range identifies the exact collection of attainable outputs Easy to understand, harder to ignore..
As an example, a transformation from (\mathbb{R}^2) to (\mathbb{R}^3) may produce only vectors lying on a particular plane or line. Even so, although its codomain is three-dimensional, its range might have dimension two, one, or even zero. This distinction is fundamental in linear algebra Worth knowing..
Definition of the Range
Let
[ T:V\rightarrow W ]
be a linear transformation from the domain (V) to the codomain (W). The range of (T), also called its image, is defined by
[ \operatorname{Range}(T)=T(V)={T(\mathbf{v}):\mathbf{v}\in V}. ]
In words, the range contains every vector (\mathbf{w}\in W) for which there is at least one vector (\mathbf{v}\in V) satisfying
[ T(\mathbf{v})=\mathbf{w}. ]
The range is always a subspace of the codomain (W). This is true because linearity guarantees that:
- The zero vector is in the range, since (T(\mathbf{0}_V)=\mathbf{0}_W).
- The sum of two outputs is also an output.
- Any scalar multiple of an output is also an output.
Range Versus Codomain
The codomain is the vector space into which a transformation is defined to map. The range is the portion of that codomain actually reached by the transformation.
Consider the transformation
[ T:\mathbb{R}^2\rightarrow\mathbb{R}^3 ]
defined by
[ T(x,y)=(x,y,0). ]
Its codomain is all of (\mathbb{R}^3), but every output has a third coordinate equal to zero. Which means,
[ \operatorname{Range}(T)={(x,y,0):x,y\in\mathbb{R}}. ]
Geometrically, the range is the (xy)-plane inside (\mathbb{R}^3), not the entire three-dimensional codomain It's one of those things that adds up. Still holds up..
This example illustrates an important rule:
The range is always contained in the codomain, but it does not necessarily equal the codomain.
If the range equals the codomain, the transformation is called onto or surjective.
Range of a Matrix Transformation
Every matrix defines a linear transformation through multiplication. Suppose (A) is an (m\times n) matrix and
[ T_A:\mathbb{R}^n\rightarrow\mathbb{R}^m ]
is defined by
[ T_A(\mathbf{x})=A\mathbf{x}. ]
If the columns of (A) are (\mathbf{a}_1,\mathbf{a}_2,\ldots,\mathbf{a}_n), then for
[ \mathbf{x}= \begin{bmatrix} x_1\ x_2\ \vdots\ x_n \end{bmatrix}, ]
the product can be written as
[ A\mathbf{x} =x_1\mathbf{a}_1+x_2\mathbf{a}_2+\cdots+x_n\mathbf{a}_n. ]
Thus, every output is a linear combination of the columns of (A). This means
[ \operatorname{Range}(T_A)=\operatorname{Col}(A). ]
The range of a matrix transformation is therefore its column space.
How to Find the Range of a Matrix Transformation
To determine the range of (T_A(\mathbf{x})=A\mathbf{x}), follow these steps:
- Identify the columns of the matrix (A
