Kernel and image of a linear transformation :
Choice of bases
for W gives the isomorphisms
(identifications) V with
and W with .
In this case the
kernel and the image of the linear transformation T coincide with the
kernel and the image of the matrix AT (see sheet #6) with respect to
the chosen bases.
Theorem. Suppose AT is the matrix of a linear operator with respect to some basis for V. Then the characteristic polynomial (see sheet ) depends only on the operator T and does not depend on the choice of basis for V.
Proof. A change of basis for V changes the matrix of the operator
T as follows
Corollary. Eigenvalues of the matrix AT (the roots of the
characteristic polynomial (see sheet ) do not depend on
the choice of basis for V. So they are invariants of the operator
Definition. An eigenvector corresponding to an eigenvalue of a linear operator T is a vector such that .
Definition. All eigenvectors corresponding to a given eigenvalue form a subspace of V which is called eigenspace corresponding to .
Theorem. Eigenvectors corresponding to distinct eigenvalues are
Suppose AT is a matrix of T with respect to some basis. Then T is diagonalizable if there is an invertible matrix P such that is a diagonal matrix.
If T has n distinct real eigenvalues, where . Then T is diagonalizable and the basis with respect to which T has a diagonal matrix consists of the n eigenvectors.