Definition of eigenspace of a linear transformation
Created almost 9 years ago, updated 5 months ago
ID24050
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).
**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).