28.The image and kernel of a linear operator.

The image (or range) of a linear operator A is the set of images of all elements , i.e. elements Ax. It is denoted by . Thus, .

Obviously, for every linear operator its domain coincides with V. The answer on the question “What is the image of a linear operator?” gives the following:

Theorem 1. Let A be a linear operator acting in a linear space V. Then

1. The set of elements is a subspace of V.

2. If V is an n-dimensional space with a basis , then the dimension of is equal to the rank of the matrix .

The rank of a linear operator A is called the dimension of its image.

Corollary 1. The rank of a linear operator A is equal to and doesn’t depend on choice of a basis .

Corollary 2. The dimension of the image of a linear operator A acting on some subspace of a linear space V doesn’t exceed . Other important characteristic of a linear operator is the collection of elements of a linear space V called the kernel of the linear operator and denoted by ker A. The kernel of a linear operator A consists of elements such that . Thus, ker . Theorem 4. If V is an n-dimensional linear space and , then ker A is a subspace of V and .

25. The coordinate representation of linear transformations. Changing the matrix of a linear transformation at a basis replacement.

Let a linear transformation A be given in a n–dimensional space of which vectors е₁, е₂, …, е_n form a basis. Since Ае₁, Ае₂, …, Ае_n are vectors of the space , then each of them can be expressed by a unique way through vectors of the basis:

Ae₁ = a₁₁e₁ + a₂₁e₂ + … + a_n1e_n,

Ae₂ = a₁₂e₁ + a₂₂e₂ + … + a_n2e_n,

………………………………..

Ae_n = a_1ne₁ + a_2ne₂ + … + a_nne_n.

The matrix is called the matrix of the linear transformation А in the basis е₁, е₂, …, е_n.

Theorem. There is a one-to-one correspondence between the set of all linear transformations of a n-dimensional linear space V and the set of all matrices of dimension

A linear transformation A in a finitely dimensional space is called regular (nonsingular) if the determinant of the matrix of this transformation differs from zero.

Every regular linear transformation A has an inverse transformation A ^{– 1} and only one.

Let and be two bases of a n-dimensional space V, and let these bases are connected by the transition matrix , i.e. for each .

Theorem. The matrix of a linear transformation in a basis is connected with the matrix of this transformation in a basis by .

26.Invariant subspaces of a linear transformation. Examples of invariant subspaces. Theorem on an eigen-subspace of a linear transformation. A subspace of a linear space is called invariant with respect to a linear transformation A if for every element x of its image Ax also belongs to . Examples: 1) The subspace consisting of one zero-element 0 is an invariant subspace with respect to any linear transformation.2) A linear space itself is invariant with respect to any linear transformation acting in this space. Zero-subspace and are called trivial invariant subspaces of a linear transformation.3) The set of radius-vectors of the points of some line on plane Oxy passing through the origin of coordinates is an invariant subspace of the operator of turning these radius vectors on angle around the axis Oz.

4) For the operator of differentiation in the linear space of functions f(t) having on the derivative of any order the linear hull of elements where are pairwise distinct constants is an n-dimensional invariant subspace. Consider now the conditions for which there exists a one-dimensional invariant subspace of a linear transformation. A non-zero vector x  V is called an eigenvector of a linear transformation А if there is such a number  that the equality Ax = x holds. The number  is called a characteristic number (eigenvalue) of the linear transformation А corresponding to the vector х.Theorem 1. The set V_ containing zero-element and all the eigenvectors of a linear transformation A corresponding to a characteristic number  is an invariant subspace of the linear transformation A. Theorem 2. If are distinct characteristic numbers of a linear transformation A then the corresponding to them eigenvectors are linearly independent.

Theorem 6. In a real n-dimensional linear space V every linear transformation has either at least one eigenvector or a 2-dimensional invariant subspace.

7. If a matrix А of a linear transformation А is symmetric then all the roots of the characteristic equation |A – E| = 0 are real numbers.

27.Characteristic numbers and eigenvectors of a linear transformation. Theorems on eigenvalues and eigenvectors.

A non-zero vector x  is called an eigenvector of a linear transformation А if there is such a number  that the equality Ax = x holds. The number  is called a characteristic number (eigenvalue) of the linear transformation А corresponding to the vector х.

Remark on importance of eigenvectors. Assume that for some linear transformation A acting in an n-dimensional linear space V n linearly independent eigenvectors have been found. It means that the following equalities hold: . Taking these elements as a basis we can conclude that the matrix of a linear transformation A in this basis will have the following diagonal form: for which studying the properties of this transformation is essentially simplified. The equation is called a characteristic equation, and – characteristic polynomial of the linear transformation A acting in V. Theorem 1. The set _ containing zero-element and all the eigenvectors of a linear transformation A corresponding to a characteristic number  is an invariant subspace of the linear transformation A.

Theorem 2. If are distinct characteristic numbers of a linear transformation A then the corresponding to them eigenvectors are linearly independent. Corollary 3. A linear transformation acting in a linear space of dimension n cannot have more than n distinct characteristic numbers.

Theorem 4. If a linear transformation acting in a linear space of dimension n has n distinct characteristic numbers then the corresponding to them eigenvectors form a basis of the space . Theorem 5. In a complex n-dimensional linear space V every linear transformation has at least one eigenvector.

Theorem 6. In a real n-dimensional linear space V every linear transformation has either at least one eigenvector or a 2-dimensional invariant subspace.

Theorem 7. If a matrix А of a linear transformation А is symmetric then all the roots of the characteristic equation |A – E| = 0 are real numbers.