Calculation rate

Course: Linear Algebra I / Matrix Calculus

Rank and associated subspaces

a) matrices as vector space (for repetition)

In addition to the elementary line operations, the following operations are obvious:

Addition of matrices of the same type:
.
Scalar multiplication of a matrix by a constant:
.

With these operations the crowd becomes to a K-vector space. Test question: What is the dimension of this vector space?

Another (single digit) operation:

Transposition of a matrix:
.

The columns and rows are swapped with one another.

Regulate: and i.e. is linear (and objectively), i.e. an isomorphism of the vector spaces.

b) subspaces associated to a matrix

A matrix we assign three subspaces:

Note: The line space of a matrix is ​​invariant under line operations (i.e. with the Gaussian algorithm).

From theorems 1.10 and 2.8 we get:

Theorem 3.1 (3rd dimensional formula) [edit]

.

Only for the field of real numbers (or its subfields like ) the following statement applies:

Lemma 3.2 [edit]

Be , then , in particular are and complementary.

We can now give a characterization of the rank that is independent of the Gaussian algorithm.

Definition 3.3 [edit]

The rank of a matrix is ​​the maximum number of linearly independent rows (or columns) of a matrix, i.e. .

Matrix product [edit]

The multiplication of matrices can be traced back to the combination of linear mappings. Idea: Be and two matrices, then the link yields a linear map that is straight from the product of the matrices should be induced, i.e. the product matrix is given by the equation introduced.

Definition 3.4 [edit]

The product of two matrices and is defined if the number of columns in A equals the number of rows in B (i.e. ) and results in a matrix of the type Number of lines (A) Number of columns (B): .

The product of matrices induces mappings .

If the products are defined, the following rules apply:

a) ,
b) ,
c) ,
d) ,
e) ,
f) , in which and the (n, n) matrix from the n unit vectors.

Attention: The product is usually not commutative even for square matrices.

Simplified notation:

a) A linear system of equations with an extended matrix of coefficients
b) The linear mapping assigned to a matrix A. .
c)
d)
e)

Usually the point is left out of the matrix multiplication.

Regular matrices [edit]

In the following section only square matrices are considered: . Here the multiplication can be carried out without restriction. We ask about the existence of one inverse matrix, i.e. after a solution of the matrix equation .

Theorem 3.5 [edit]

The matrix equation , , is solvable iff. . If the equation is solvable, then the solution is unique.

We denote the clear solution with who have favourited the inverse of . We call square matrices with maximum rank regular. We denote the set of all regular matrices with .

Definition 3.6 [edit]

A matrix means regular if .

Calculation rules:

Be , than are and also regular and the following applies:
and .

Test questions:

What does the reduced row-step form of a regular matrix look like?
What rank has ? (Reason?)

Calculation method: determination of the inverse

Transfer in the reduced line-step form. Result: .

Remarks: The set of all regular matrices forms a group, the linear group. We will only introduce the group concept later, here this means: The product of two regular matrices is regular again, and so is their inverse. Description: .

Elementary matrices [edit]

The elementary row (or column) operations are induced by multiplication with the so-called elementary matrices.

Definition 3.7 [edit]

The use of an elementary line operation , or on the identity matrix results in a corresponding elementary matrix, which we use according to the elementary operation with , or. describe.

Theorem 3.8 [edit]

Be one of the elementary line operations , , and