Definition: The pre-image of a subset \(B \subseteq W\) under a linear transformation \(T: V \to W\) 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.
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