LINEAR ALGEBRA (Math 211)
Sheet #8
LINEAR TRANSFORMATIONS 2

Kernel and image of a linear transformation :

Choice of bases , , ..., for V and , , ..., 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 . Then

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 T.

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 linearly independent.

Diagonalization. A linear operator is diagonalizable if there is a basis for V with respect to which the matrix of T is diagonal.

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.