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Definition of range of a linear transformation

Created over 8 years ago, updated 10 days ago

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).