**Definition:** Let \(T: V 	o V\) be a linear transformation and \(\lambda\) an eigenvalue of \(T\). The *eigenspace* corresponding to \(\lambda\) is:
\[E_\lambda = \{\mathbf{v} \in V : T(\mathbf{v}) = \lambda \mathbf{v}\} = \ker(T - \lambda I)\]

The eigenspace is the set of all eigenvectors corresponding to \(\lambda\), together with the zero vector.

**Example:** For \(T(x,y,z) = (2x, 2y, 3z)\), the eigenspace for \(\lambda = 2\) is \(E_2 = \{(x, y, 0) : x, y \in \mathbb{R}\}\) (the xy-plane), and \(E_3 = \{(0, 0, z) : z \in \mathbb{R}\}\) (the z-axis).