Linear Algebra: Chapter 3 linear equations 1

§ 1 Elimination Method

I. Elementary Transformation of Linear Equations

Now we will discuss general linear equations. The so-called general linear equations refer to the form

(1)

Equations, which represent an unknown number, are the number of equations, called the coefficients of linear equations, known as constant items. the number of unknowns in the equations is not necessarily equal to the number of equations. the first indicator of the coefficient indicates that it is in the first equation, and the second indicator indicates that it is a coefficient.

A solution of the so-called equations (1) refers to an ordered array composed of numbers. When each equation in (1) is transformed into an equality. the whole solution of equations (1) is called its solution set. solving a equations is actually finding out all its solutions, or finding a set of its solutions. if two equations have the same set of solutions, they are called the same solution.

Obviously, if we know all the coefficients and constant terms of a linear equations, the linear equations are basically determined. To be exact, the linear equations (1) can use the following matrix

(2)

. In fact, with (2), the linear equations except words that represent unknown numbers (1) are determined, and the words used to represent unknown numbers are of course not substantive. I learned how to use addition and subtraction elimination methods and substitution elimination methods to solve binary and ternary linear equations in the Algebra I learned in middle school. in fact, this method is more universal than solving linear equations with a determinant. the following describes how to use the general elimination method to solve general linear equations.

For example, solving equations

The second equations subtract twice the first equation, and the third equation minus the first equation becomes

The second equation is two times the third equation, and the order of the second and third equations is interchangeable.

In this way, it is easy to obtain the solution of the equations (9,-1,-6 ).

By analyzing the elimination method, it is not difficult to see that it is actually repeatedly transforming the equations, and the transformations used are only composed of the following three basic transformations:

1. Use a non-zero number to multiply an equation;

2. Add a multiple of one equation to another;

3. interchange the positions of two equations.

Definition 1Transformation 1, 2, 3 is called the elementary transformation of linear equations.

Ii. Solutions of Linear Equations

The elementary transformation process is the process of repeating the elementary transformation. It is proved that the elementary transformation always converts the equations into the equations of the same solution.

The following describes how to use elementary transformations to solve a general linear equations.

For equations (1), first check the coefficient. if the coefficients are all zero, then equations (1) can take any value without any restrictions, and equations (1) can be considered as equations. if the coefficient is not all zero, you can use elementary transformation 3. use Elementary Transformation 2 to add the times of the first equation to the first equation (). so the equations (1) become

(3)

Where

In this way, the problem of solving the equation group (1) comes down to solving the equation group.

(4)

. Obviously, a solution of (4) is the value defined by the first equation of (3), and a solution of (3) is obtained. (3) the solutions are obviously (4. that is to say, equations (3) must have sufficient conditions for solutions: equations (4) have solutions, and (3) and (1) are the same solutions, because equations (1) the sufficient and necessary conditions for solutions are equations (4.

(4) then, according to the above considerations, and step by step, finally, we will get a type ladder equations. To facilitate the discussion, we may wish to set the equations

(5)

Where. some equations such as "0 = 0" in equations (5) may not or may appear. Removing them does not affect the solution of (5. and (1) and (5) are the same solutions.

Now consider the solution of (5.

For example, (5) There is an equation, and no matter what value is taken, it cannot be an equation. Therefore (5) There is no solution, so (1) there is no solution.

When there is no "0 = 0" equation in zero or (5), there are two situations:

1). In this case, the factorial equations are

(6)

The values starting from the last equation can be uniquely determined one by one. In this case, equations (6), that is, equations (1) have a unique solution.

Example 1Solving Linear Equations

2). In this case, the factorial equations are

Here, rewrite it

(7)

It can be seen that given a group of values is a unique value, that is, a solution to the equations (7. generally, from (7), we can express the pass. Such a group of expressions is called the general solution of equations (1) and a group of free unknown numbers.

Example 2Solving Linear Equations

From this example, we can see that the general linear equations are not necessarily the form of (5), but as long as some items in the equations are transferred, they can always be the form of (5.

The above is the whole process of solving linear equations using elimination method. in general, the first step is to use the elementary transformed linear equations as the step ladder equations and remove some final equations "0 = 0" (if any. if the last equation in the remaining equation is zero or zero, there is no solution to the equations. Otherwise, there is a solution. if there is a solution, if the number of equations in the factorial equations is equal to the number of unknowns, then the equations have a unique solution. If the number of equations in the stepped equations is less than the number of unknowns, then the equations have an infinite number of solutions.

Theorem 1In homogeneous linear equations

If so, it must have a non-zero solution.

Matrix

(10)

It is called the Augmented Matrix of linear equations (1. obviously, using elementary transformed equations (1) to form a step ladder is equivalent to using Elementary Line transformed Augmented Matrix (10) to form a step ladder matrix. therefore, the first step for solving linear equations can be carried out through matrices. The step-by-step matrix can determine whether the equations have or do not have solutions. In the case of solutions, return to the step-by-step equations.

Example 3Solving Linear Equations

 

 

 

 

 

 

 

 

 

§ 2-dimensional Vector Space

Definition 2A dimension vector on a number field is an ordered array composed of the numbers in the number field.

(1)

It is called the component of vector (1.

Uses lowercase Greek letters to represent vectors.

Definition 3If the dimension vector

The corresponding components of are equal, that is

.

The two vectors are equal.

The basic relationship between dimension vectors is expressed by Vector Addition and number multiplication.

Definition 4Vector

Called Vector

And is recorded

Launched immediately by definition:

Exchange Law:. (2)

Combination law:. (3)

Definition 5Zero-Weight Vector

It is called a zero vector and recorded as 0. A vector is called a negative vector of a vector and recorded.

Obviously, all

. (4)

. (5)

(2)-(5) is the four basic calculation rules of Vector Addition.

Definition 6

Definition 7Set as number in the number field, Vector

The product of the quantity of a vector and a number.

Launched immediately by definition:

, (6)

, (7)

, (8)

. (9)

(6)-(9) is the four basic calculation rules for multiplication of numbers. It is not difficult to introduce from (6)-(9) or by definition:

, (10)

, (11)

. (12)

If, then

. (13)

Definition 8We take the number in the number field as the whole of the dimension vector of the component, and take into account the addition and quantity multiplication defined above them, which is called the dimension vector space in the number field.

In this case, the three-dimensional vector space can be considered as the space of all vectors in the geometric space.

In the preceding section, the set of all dimension vectors in the number field is composed of an addition method and a number multiplication Algebra Structure, that is, the dimension vector space in the number field.

A vector is usually written as a line:

.

You can also write a column:

.

For difference, the former is called a row vector, and the latter is called a column vector. They differ only in writing.

 

 

§ 3 linear correlation

In general, except for the zero space composed of only one zero vector, vector space contains an infinite number of vectors. The relationship between these vectors is crucial to understanding the structure of vector space.

1. linear correlation is not linear.

The simplest relationship between two vectors is proportional. A vector is proportional to a number.

.

Definition 9A vector is called a linear combination of vector groups. If there is a number in a number field

,

It is called the coefficient of this linear combination.

For example, any dimension vector is a vector group.

(1)

A linear combination.

A vector is called a dimension unit vector.

Zero vectors are linear combinations of arbitrary vector groups.

When a vector is a linear combination of vector groups, it can also be listed in a linear form of vector groups.

Definition 10If every vector in a vector group can be listed in a linear form of a vector group, the vector group can be listed in a linear form of a vector group. If two vector groups can be listed in a linear form, they are called equivalent.

By definition, each vector group can be listed in its own linear form. at the same time, if a vector group can be listed in linear form by a vector group and a vector group can be listed in linear form by a vector group, then a vector group can be listed in linear form by a vector group.

Vector groups are equivalent to the following:

1) inversion: each vector group is equivalent to itself.

2) symmetry: If the vector group is equivalent, the vector group is equivalent.

3) transmission: If vector groups are equivalent to equivalence, then vector groups are equivalent.

Definition 11If a vector in a vector group can be listed in a linear form of other vectors, the vector group is linearly related.

From the definition, we can see that any vector group containing zero vectors must be linearly related. linear Correlation of vector groups indicates or (these two formulas may not be supported at the same time ). when it is a real number field and three-dimensional, it indicates the vector and the common line. the geometric significance of linear correlation between the three vectors is that they are in common.

Define 11 ′Vector groups are called linear correlation. If there is a number that is not completely zero in the number field

These two definitions are consistent.

Definition 12Always the number group is not linearly related, that is, there is no incomplete zero number, so

It is called linear independence; or, a constant group is called linear independence.

Available

It is defined as linear correlation. in other words, if the linear correlation is always the same as that of a group, any non-empty group is also Linearly Independent. in particular, because two proportional vectors are linearly related, linear unrelated vector groups cannot contain two proportional vectors.

When 11' is defined to contain a vector group composed of a vector group, a single zero vector is linearly correlated and a non-zero vector is linearly independent.

It is not hard to see that vector groups composed of dimension unit vectors are linearly independent.

The problem of determining whether a vector group is linear or linear independent can be attributed to solving the equations.

(2)

Linear Correlation. According to definition 11, the equation is used.

(3)

Whether there is a non-zero solution. (3) The formula is written by component.

(4)

As a result, the sufficient and sufficient condition for Linear Independence of vector groups is that the homogeneous linear equations (4) have only zero solutions.

Example 1Judgment Vector

Linear correlation.

Example 2In vector space, for any non-negative integer

Linear Independence.

Example 3If the vector group is linearly independent, the vector group is also Linearly Independent.

Therefore, if vector group (2) is linearly independent, then add a component to each vector to obtain the dimension vector group.

(5)

Linear Independence.

Theorem 2It is set to two vector groups. If

1) vector groups can be listed in linear form,

2 ),

Then the vector group must be linearly related.

Inference 1If a vector group can be linearly expressed by a vector group and has no linear relationship.

Inference 2Any dimension vector must be linearly related.

Inference 3Two Linear unrelated vector groups must contain the same number of vectors.

The geometric meaning of Theorem 2 is clear: in the case of a three-dimensional vector, if, then the vectors that can be listed by the vector linear form are of course on the plane where they are located, so these vectors are co-faces, that is to say, at that time, these vectors were linearly related. two vector groups are equivalent to each other, which means they are on the same plane.

Ii. Massively linear independent group

Definition 13A part of a constant group is called a very large linear independent group. If the group itself is linear independent and a vector (if any) is added to the vector group ), some vector groups are linearly correlated.

The maximum linear independent group of a Linear Independent Vector Group is the vector group itself.

A basic property of a strongly linear independent group is that any extremely large linear independent group is equivalent to the vector group itself.

Example 4Vector Group

Here {} is linearly independent, so {} is a strongly linear independent group. On the other hand, {} and {} are also extremely linear independent groups of vector groups.

The preceding example shows that the maximum linear independent group of a vector group is not unique. however, each extremely large linear independent group is equivalent to the vector group itself. Therefore, any two extremely large linear independent groups of a constant group are equivalent.

Theorem 3The maximum linear independent group of a constant group contains the same number of vectors.

Theorem 3 indicates that the number of vectors contained in a maximum linear independent group is irrelevant to the choice of a maximum linear independent group. It directly reflects the nature of the vector group.

Definition 14The number of vectors contained in the maximum linear independent group of a vector group is called the rank of this vector group.

A sufficient condition for linear independence of a mass group is that its rank is the same as the number of vectors it contains.

Each constant group is equivalent to its extremely linear independent group. it can be seen from the equivalence pass that the maximum linear independence group of any two equivalent vector groups is also equivalent. therefore, an equivalent vector group must have the same rank.

Vector groups containing non-zero vectors must have extremely large linear independent groups, and any linear independent vector can be expanded into a large linear independent group. all vector groups composed of zero vectors do not have extremely large linear independent groups. specify that the rank of such a vector group is zero.

Now we associate the above concepts with the solutions of the equations, and give a Equations

The vectors corresponding to each equation are

There is another equation.

The corresponding vector is. then linear combination. if and only if, that is, equation (B) is a linear combination of equations. it is easy to verify that the solution of the equations must satisfy (B ). further setup of Equations

The vector corresponding to its equation is. If it can be listed in a linear form, the solution of the equations is the solution of the equations. Further, when it is equivalent, the two equations are the same solution.

Example 5(1) is the above proposition true for a Linear Independent Vector Group?

(2) Is it linear when it proves that it is also linear?

Example 6It is located in a vector group, and each one cannot be expressed as a linear combination of its first vector, proving linear independence.