Definition: The range of a linear transformation (T: V o W) is the set of all output vectors: [ ext{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 ext{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 ext{range}(T)), and (c\mathbf{w}_1 = cT(\mathbf{v}_1) = T(c\mathbf{v}_1) \in ext{range}(T)).
The dimension of the range is called the rank of (T).
**Definition:** The *range* of a linear transformation \(T: V o W\) is the set of all output vectors:
\[ ext{range}(T) = \{T(\mathbf{v}) : \mathbf{v} \in V\} \subseteq W\]
**Theorem:** The range of \(T\) is a subspace of \(W\).
**Proof sketch:** If \(\mathbf{w}_1, \mathbf{w}_2 \in ext{range}(T)\), 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 \(\mathbf{w}_1 + \mathbf{w}_2 = T(\mathbf{v}_1) + T(\mathbf{v}_2) = T(\mathbf{v}_1 + \mathbf{v}_2) \in ext{range}(T)\), and \(c\mathbf{w}_1 = cT(\mathbf{v}_1) = T(c\mathbf{v}_1) \in ext{range}(T)\).
The dimension of the range is called the *rank* of \(T\).