- •Homogeneous systems of linear algebraic equations.
- •The inverse of a matrix. Matrix equations.
- •If a matrix a has an inverse, then a is invertible (or non-singular); otherwise, a is singular. A nonsquare matrix cannot have an inverse. Not all square matrices possess an inverse.
- •Definition and examples.
- •Linear independence and basis.
- •Linear transformations.
- •If two matrices are connected with such a correlation they are called similar. The determinants of similar matrices are equal.
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:
identity transformation, that is ;
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 :
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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
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Solution:
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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.