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.
**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.