Matrix Theory 1 linear space

Why Learning matrix theory? How to learn and master?

• Vector, matrix and its algorithm are powerful tools used to describe, analyze, and process linear systems. The powerful tool is embodied in its universality and simplicity.
  
• Application of basic knowledge in further Cognitive Science Research in professional courses
  

 

First, linear space

 

1. Definitions and Properties of Linear Spaces

[Prerequisites]

★Set: in general, it refers to the whole composed of some things (or objects, called elements.

Expression of the set: enumeration and expression, such

;;

Set operation: sum (), intersection ()

In addition, the "and" () of a set is not a set operation in a strict sense, because it limits the addition of elements in the set.

★Number Field: A number set that is closed to the four arithmetic operations (the divisor is not zero ). Such as rational number field, real number field (), and complex number field (). Real number field and complex number field are commonly used in engineering.

Linear Space is one of the most basic concepts of linear algebra and an important basis for learning matrix theory. The concept of linear space is a unified abstraction of various specific linear systems.

 

1. Linear Space definition:

It is a non-empty set, and its elements are represented by equals; it is a number field, and its elements are represented by equals. If

(I) Define an "addition" Operation in, that is, there is a unique "and" (closed) at that time, and the addition operation satisfies the following properties:

(1) Combination law;

(2) Exchange Law;

(3) The zero-Element Law has zero elements;

(4) Negative metabases have one element for any element, which is also known as a negative element. That is ,.

(II) define a "multiplication" Operation in, that is, when, there is a unique "product" (closed), and the multiplication operation satisfies the following properties:

(5) number Factor Allocation law;

(6) allocation law;

(7) combination law;

(8) constant law;

It is called the linear space in the number field.

Note:

(1) linear space cannot be defined by leaving a certain number field. In fact, for different number fields, the linear space of the same set may be different. One can even be a linear space, and the other cannot be a linear space.

(2) Two operations and eight properties

The operation in the number field is a specific arithmetic operation, while the addition operation and multiplication operation defined in it can be very abstract.

(3) In addition to two operations and eight properties, the uniqueness and closeness should also be paid attention.

The uniqueness is generally obvious, and the closeness also needs to be proved. The case is not closed: the set is small, and the operation itself is not satisfied.

When the number field is a real number field, it is called a real linear space; when it is a complex number field, it is called a complex linear space.

 

Example 1: Set = {all positive numbers}, and its "addition" and "number multiplication" operations are defined

Xy=Xy,

Proof: It is a real number fieldRLinear Space.

Proof: first, we need to prove the uniqueness and closeness of the two operations.

① Uniqueness and closeness

Uniqueness

IfX> 0,Y> 0, Then there is

Xy=XyClosed proof.

② Eight properties

(1)X(Yz) =X(Yz) = (Xy)Z= (Xy)Z

(2)Xy=Xy = yx

(3) 1 is a zero element.X1 =
[XO = x --> XO = x-> O = 1]

(4) YesXNegative elementX= [X + y = O]

(5 )(XY)XY[Number Factor Allocation law]

(6 )(X)(X) [Allocation law]

(7) [combination law]

(8) [constant law]

This proves that it is a real number field.RLinear Space.

 

1. Theorem: linear space has the following properties:
   
   1. The zero element is unique, and the negative element of any element is unique.
      
   2. The following equality is true: o ,.
      

Proof: (1) using the reverse verification method:

① The zero element is unique. If there are two zero elements O1 and O2, because both O1 and O2 are zero elements, there is a [exchange law]

O1 + O2 = O1 = O2 + O1 = O2

So O1 = O2

That is, O1 and O2 are the same, which is in conflict with assumptions. Therefore, there is only one zero element.

② The negative element of any element is also unique. Assume that two negative elements and

O =

[Zero-element law] [combination law] [zero-Element Law]

That is, it is the same as, so the negative element is unique.

(2) ①: SetW= 0X, ThenX+W= 1X+ 0X= (1 + 0)X=X, HenceW= O.

[Constant law]

②: SetW= (-1)X, ThenX+W= 1X+ (-1)X= [1 + (-1)]X= 0X= O, soW=-X.

 

1. Linear Correlation
   

The linear correlation concept in linear space is similar to that in linear algebra.

• Linear combination:

A linear combination of element groups.

• Linear representation: an element inXIt can be expressed as a linear combination of one of the element groups.XIt can be linearly represented by this element group.

• Linear correlation: if there is a set of incomplete zero numbers, the element has

It is called linear correlation of element groups, otherwise it is called linear independence. [Linear correlation is a very important concept. with linear correlation, we can see the dimensions, bases, and coordinates of the following linear spaces]

1. Dimension of Linear Space
   

Definition: the dimension of the number of elements contained in the maximum linear independent element group in a linear space.

This course only takes into account limited dimensions and does not cover unlimited dimensions.

 

Example 2. AllM×NThe set of real-order matrices forms a real linear space (for matrix addition and number multiplication operations on the matrix), and its dimension is obtained.

Solution: a direct method is to find a maximum linear independent group, and its elements are as simple as possible.

LingEijFor suchM×NLevel matrix, which (I, j) The element is 1, and the remaining elements are zero.

Obviously, such a matrix has a totalMn, ConstituteMnLinear independent element group. On the other hand, the maximum number of elements is required. Any element group can be linearly represented by the preceding element group,

-->

That is, the largest linear independent element group is formed, so the dimension of the space isMn.

 

1. Basis and coordinates of Linear Space
   
   1. Definition of the base: SetVYesfieldKThe linear space on isVOfRAny element.
      

(1) linear independence;

(2)VAny vector inXCan be linearly expressed.

It is calledVIs called the base element of the base.

• Kizheng isVMaximum Linear Independent element group;VThe dimension is the number of elements contained in the base.

• The base is not unique, but the number of elements contained in different bases is equal.

 

1. Consider the set of all plural numbersC. IfK = C(Complex number field), the set forms a linear space for multiplication of the plural addition and the plural number. The base can be 1, and the space dimension is 1.K = R(Real number field), then this set forms a linear space for the Number Addition of the complex number and the number multiplication of the real number to the complex number. Its base can be {1,I}, The space dimension is 2.
   

 

Number FieldK	Two types of operations	Base	General Element	Space type	Dimension
Complex FieldC	(1) plural addition; (2) multiplication of plural numbers	{1}		Complex Linear Space	1
Real Number FieldR	(1) plural addition; (2) real number logarithm of the plural number multiplication	{1,I}		Real Linear Space	2

1. Coordinate definition: a coordinate system based on a linear space. Its linear representation under this base is:
   

It is calledXCoordinates or components in the coordinate system are recorded

(1) In general, linear spaces and their elements are abstract objects. elements in different spaces can have different categories and properties. However, the coordinate representation unifies them. The coordinate representation leaves the difference to the base and base elements. The new vector composed of coordinates is only represented by the number in the number field.

(2) Furthermore, the original abstract "addition" and "multiplication" evolved into vector addition and number-to-vector multiplication after coordinate representation.

Positive correspondence

Positive correspondence

(3) Obviously, the coordinates of the same element in different coordinate systems are different. We will also study this transformation relationship later.

 

1. Base transformation and Coordinate Transformation
   

The basis is not unique. Therefore, we need to study the law of coordinate transformation when the basis changes.

The old base is set, and the new base is set. because both are bases, they can be linearly expressed to each other.

()

That is

WhereCCalled the transition matrix, the above formula provides the basis transformation relationship, which can prove that,CIs reversible.

It is linearly expressed

It is linearly represented

Then

The coordinate transformation relationship is obtained due to linear independence of the base element.

 

 

Job: P25-26

Supplement: It is proved that for zero element o in linear spaceKO = o.