Definition of pre-image (of a point) under a linear transformation

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}).