In linear algebra the term "linear operator" most commonly refers to linear
maps (i.e., functions
preserving vector addition and scalar multiplication) that have the added
peculiarity of mapping a vector space into itself (i.e.,
).
The term may be used with a different meaning in other branches of
mathematics.
Before providing a definition of linear operator, we need to remember that a
function
that associates one and only one element of a vector space
to each element of another vector space
is said to be a linear map if and
only
if
for
any two scalars
and
and any two vectors
.
Linear operators are defined analogously.
Definition
Let
be a vector space. A function
is said to be a linear operator if and only
if
for
any two scalars
and
and any two vectors
.
Let us provide a simple example.
Example
Consider the space
of all
column vectors having real entries. Suppose the function
associates to each vector
a
vector
Choose
any two vectors
and any two scalars
and
.
By repeatedly applying the definitions of
vector addition and
scalar
multiplication, we
obtain
Therefore,
is a linear operator.
Since a linear operator is a special kind of linear map, it inherits all the properties of linear maps. For convenience, we report here the most important of these inherited properties, but if you are already familiar with linear maps, you can safely skip this section.
A linear operator
is completely determined by its values on a
basis of
.
Proposition
Let
be a linear space,
a basis for
,
and
a set of
elements of
.
Then, there is a unique linear operator
such
that
for
.
See the proof in the lecture on linear maps.
Multiplication of vectors by a square matrix defines a linear operator.
Proposition
Let
be the linear space of all
column vectors. Let
be a
matrix. Let
be defined, for any
,
by
where
denotes the matrix
product between
and
.
Then
is a linear operator.
See the proof provided in the lecture on linear maps.
Proposition
Let
be the linear space of all
row vectors. Let
be a
matrix. Consider the transformation
defined, for any
,
by
where
denotes the matrix product between
and
.
Then
is a linear operator.
As before, see the proof in the lecture on linear maps.
The "linearity preserving" property extends to linear combinations involving more than two terms.
Proposition
Let
be
scalars and let
be
elements of a linear space
.
If
is a linear operator,
then
Also in this case, see the proof in the lecture on linear maps.
Remember that every linear map
between two finite-dimensional vector spaces can be represented by a matrix
,
called the matrix of the
linear map. The notation
indicates that the matrix depends on the choice of two
bases: a basis
for the space
and a basis
for the space
.
The matrix is constructed as follows:
where
the columns are the coordinate
vectors of the transformations
of the vectors belonging to the basis
.
The number of columns of
is equal to the number of elements in the basis
,
while the number of rows of
is equal to the number of elements in the basis
.
In the case of a linear operator, the codomain
coincides with the domain
,
that is,
.
There are two important consequences of this fact.
First, any two bases
and
of
have the same number of elements (by the
dimension theorem).
Therefore, the matrix
of a linear operator is square. Hence, we can apply to linear operators the
rich set of theoretical tools that can be applied exclusively to square
matrices (e.g., the concepts of
inverse,
trace,
determinant,
eigenvalues and
eigenvectors).
Second, we can (although we are not obliged to) use a unique basis
for both the domain and codomain. When we choose this kind of simplification,
the matrix of the linear map is
which
we can also simply denote by
.
Example
Let
be a linear space spanned by the basis
.
Suppose
is a linear operator such
that
Then,
the coordinate vectors needed to form the matrix of the linear operator
are
and
Thus,
the matrix of the linear operator with respect to
is the square
matrix
Below you can find some exercises with explained solutions.
Let
be a linear space spanned by the
basis
.
Suppose
is a linear operator such
that
Find the matrix
of the linear operator
.
After applying the linear operator, the
coordinate vectors of the elements of the basis
becomeand
and
Thus,
the matrix of the linear operator with respect to
is the square
matrix
Use the matrix
found in the previous exercise to compute how the operator
transforms the coordinates of the vector
such
that
The transformation can be computed by
performing a matrix
multiplication:
Please cite as:
Taboga, Marco (2021). "Linear operator", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/linear-operator.
Most of the learning materials found on this website are now available in a traditional textbook format.