Definition: The pre-image (or inverse image) of a vector \(\mathbf{w} \in W\) under a linear transformation \(T: V \to W\) is the set of all vectors in \(V\) that map to \(\mathbf{w}\):
\[T^{-1}(\mathbf{w}) = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{w}\}\]
The pre-image may contain zero, one, or many vectors.
Key fact: If \(\mathbf{v}_p\) is one particular solution (i.e., \(T(\mathbf{v}_p) = \mathbf{w}\)), then the complete pre-image is:
\[T^{-1}(\mathbf{w}) = \mathbf{v}_p + \ker(T) = \{\mathbf{v}_p + \mathbf{k} : \mathbf{k} \in \ker(T)\}\]
Example: For \(T: \mathbb{R}^2 \to \mathbb{R}\) defined by \(T(x,y) = x + y\), the pre-image of \(3\) is the line \(\{(x, y) : x + y = 3\}\).
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