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 *A*_{T} (see sheet #6) with respect to
the chosen bases.

**Theorem.** *Suppose A_{T} 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 A_{T} (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.*

Suppose *A*_{T} 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.