Dashboard

Now you are in the subtree of Container for Linear Algebra project. 

Definition of range of linear transformation

Created over 8 years ago, updated 10 days ago

Definition: The range (or image) of a linear transformation (T: V \to W) is the set of all possible output vectors:
[\text{range}(T) = \text{im}(T) = {T(\mathbf{v}) : \mathbf{v} \in V} \subseteq W]

The range is always a subspace of the codomain (W).

Example: For (T: \mathbb{R}^3 \to \mathbb{R}^3) defined by (T(x,y,z) = (x,y,0)), the range is the (xy)-plane: ({(x,y,0) : x,y \in \mathbb{R}}).

Example: For (T: \mathbb{R}^2 \to \mathbb{R}^2) defined by (T(x,y) = (x+y, 2x+2y)), the range is the line spanned by ((1,2)).

Dimension of the range: The dimension of the range is called the rank of (T). For a matrix transformation (T(\mathbf{x}) = A\mathbf{x}), the rank equals the number of pivot columns (or the column rank of (A)).