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Definition of pre-image of linear transformation

Created over 8 years ago, updated 10 days ago

Definition: The pre-image of a subset (B \subseteq W) under a linear transformation (T: V o W) is:
[T^{-1}(B) = {\mathbf{v} \in V : T(\mathbf{v}) \in B}]

When (B = {\mathbf{w}}) is a single vector, the pre-image is:
[T^{-1}(\mathbf{w}) = {\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{w}}]

Key property: If (B) is a subspace of (W), then (T^{-1}(B)) is a subspace of (V).

Key property: If (T(\mathbf{v}_p) = \mathbf{w}) for some particular (\mathbf{v}_p), then:
[T^{-1}(\mathbf{w}) = \mathbf{v}_p + \ker(T)]
The pre-image is a coset (translate) of the kernel.