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) 
Only for the field of real numbers (or its subfields like ) the following statement applies:
Lemma 3.2 
- 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 
- The rank of a matrix is the maximum number of linearly independent rows (or columns) of a matrix, i.e. .
Matrix product 
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 
- 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.
- a) A linear system of equations with an extended matrix of coefficients
- b) The linear mapping assigned to a matrix A. .
Usually the point is left out of the matrix multiplication.
Regular matrices 
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 
- 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 
- A matrix means regular if .
- Be , than are and also regular and the following applies:
- and .
- 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 
The elementary row (or column) operations are induced by multiplication with the so-called elementary matrices.
Definition 3.7 
- 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 
- Be one of the elementary line operations , , and
- Would you sell your voting rights
- How is the hostel in SGSITS Indore
- Machine learning workshop provider in Hyderabad
- What sports are popular in Laddakh
- What is a Smart Textile E Textiles
- What made you live a lonely life
- You can shave yourself if you have mollusks
- What science is involved in software development
- How much is my personal data worth
- What is the range of tan X 2cot X.
- How were the first Clovis points discovered
- Why did North Korea arrest Warmbier
- When does staining usually occur?
- How do I automate with Python
- Can I use Linux on my Mac?
- The multiplayer video game dies
- How do I get more alone time
- What are some good evening makeup tutorials
- Do you really like white noise
- What is hash or hashish
- When will Willie Nelson stop smoking marijuana?
- What is an automatic weather station
- What will happen if China attacks Israel?
- Is culturally thematized make-up cultural appropriation
- Is Sam's Club better than Wal Mart
- What is unusual about Venus' orbit?
- What is the timing of KVPY
- Smoking weed causes a major disease
- Which is the best mobile comparison site
- What is Superman Distant Fires
- Why is Google Webmaster doing Keyword Deindexing
- Why is my YouTube channel not running?