Definition of range of linear transformation

**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 \(A\)).Copy