# 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*

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