**Definition:** The *image* of a vector \(\mathbf{v}\) under a linear transformation \(T: V \to W\) is the output vector \(T(\mathbf{v}) \in W\).

Notation: \(T(\mathbf{v})\) or \(\mathbf{w} = T(\mathbf{v})\).

This is the result of applying the transformation to a specific input vector.

**Example:** If \(T: \mathbb{R}^2 \to \mathbb{R}^2\) is defined by \(T(x,y) = (2x, y)\), then the image of \((3,4)\) under \(T\) is \(T(3,4) = (6,4)\).