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  • Linear transformations.

Let a finite-dimensional linear space L be given. The representation is called a transformation of the linear space.

The transformation is called a linear transformation if . This means that a linear transformation transform any linear combination of vectors in a linear combination of the images of these vectors, moreover, with the same coefficients.

The following properties of linear transformations are true. , .

Let us show some examples:

  1. identity transformation, that is ;

  2. zero transformation, that is .

Let be a basis in the linear space L.

Theorem 1. For an ordered system of vectors there exists only one transformation , such that .

Thus, we have one-to-one correspondence between all linear transformations and all ordered systems of vectors . However every vector from this system can be represented as a linear combination of the basis vectors :

.

From the components of the vectors in the basis we can form the square matrix . Thus, we have one-to-one correspondence between all linear transformations and all square matrices of the order n, this correspondence, certainly, depends on the choice of the basis .

We will say that the matrix A defines the linear transformation or A is the matrix of the linear transformation in the basis . If we define as the images column of the basis, then and if , then . The linear transformation is called non-singular if it is a surjection. The matrix of a non-singular linear transformation is non-singular (i.e. its determinant is not equal to zero).

Example 45: let be the basis of linear space, matrix A be the matrix of linear transformation . Find the image of the element , if

.

Solution:

.

Let two bases and , with the transition matrix T ( ) be given. And let the linear transformation in these bases be defined by matrices and ( and . We have , but , because is the linear transformation. Thus, , and because e is the linear independent system . Finally, because T is invertible,

and .

If two matrices are connected with such a correlation they are called similar. The determinants of similar matrices are equal.