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# Definition of range of a linear transformation

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

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