Definition of range of a linear transformation

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

**Theorem:** The range of \(T\) is a subspace of \(W\).

**Proof sketch:** If \(\mathbf{w}_1, \mathbf{w}_2 \in \text{range}(T)\), then \(\mathbf{w}_1 = T(\mathbf{v}_1)\) and \(\mathbf{w}_2 = T(\mathbf{v}_2)\) for some \(\mathbf{v}_1, \mathbf{v}_2 \in V\). Then \(\mathbf{w}_1 + \mathbf{w}_2 = T(\mathbf{v}_1) + T(\mathbf{v}_2) = T(\mathbf{v}_1 + \mathbf{v}_2) \in \text{range}(T)\), and \(c\mathbf{w}_1 = cT(\mathbf{v}_1) = T(c\mathbf{v}_1) \in \text{range}(T)\).

The dimension of the range is called the *rank* of \(T\).Copy