The coordinate representation of linear transformations

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. The columns of the matrix are composed from coefficients in formulas of transformation of the basis vectors. Consider an arbitrary vector х = х₁e₁ + х₂e₂ + … + х_ne_n in the space . Since Ax  , then the vector Ах can be expressed through the basis vectors:

The coordinates of the vectorАх are expressed through coordinates (х₁, х₂, …, х_n) of the vector х by the formulas:

= a₁₁x₁ + a₁₂x₂ + … + a_1n x_n,

= a₂₁x₁ + a₂₂x₂ + … + a_2n x_n,

………………………………..

= a_n1x₁ + a_n2x₂ + … + a_nn x_n.

We can name these n equalities as a linear transformation А in the basis е₁, е₂, …, е_n. The coefficients in formulas of this linear transformation are elements of rows of the matrix .

In matrix form: ;.

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 .

Proof. It has been above showed that for every linear transformation in V is corresponded the matrix of dimension . On other hand, the expressioncan be accepted as a definition of some transformation. Its linearity follows from the rules of operations with matrices.

Example. Find the matrix of an identity transformation E in a n-dimensional space.

Solution: An identity transformation does not change the basis vectors: , , , …, , i.e.

,

,

…………………………

.

Consequently, the matrix of an identity transformation is identity matrix:

.

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.

If a regular linear transformation A in the coordinate form is determined by the following equalities:

…………………………

then the inverse linear transformation A ^{– 1} is determined as follows:

………………………………

Here is the cofactor of the elementof the matrixA, |A| is the determinant of the matrix A.

The matrix of the linear transformation A ^{– 1} is inverse according to the matrix A and is determined by the equality: .

Example. Let A_k be a transformation of turning every vector on angle in the spaceV₂ of vectors on a plane. Find matrix of the following transformations: 1) A₁A₂; 2) .

Solution: 1) The matrix of the transformation A_k is the following:

The matrix of the product of transformations is equal to the product of matrices of these transformations:

.

2) The matrix of inverse transformation is the inverse matrix forA₁:

.

Changing the matrix of a linear transformation at a basis replacement

Understand how the matrix of a linear transformation is changed at a basis replacement. Let andbe two bases of an-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 basisis connected with the matrix of this transformationin a basisby.

Proof. Let A be a linear transformation. Take an arbitrary . Then there issuch that. Let,,,

.

We have and.

Since every transition matrix has the inverse matrix, we obtain

. Consequently, .