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 iffor 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 iffor 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 vectorChoose any two vectors and any two scalars and . By repeatedly applying the definitions of vector addition and scalar multiplication, we obtainTherefore, 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 thatfor .

Proof

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.

Proof

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.

Proof

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

Proof

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 thatThen, the coordinate vectors needed to form the matrix of the linear operator areandThus, 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 .

Solution

After applying the linear operator, the coordinate vectors of the elements of the basis becomeandandThus, 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

Solution

The transformation can be computed by performing a matrix multiplication:

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