Linear algebra: Fourth chapter vector Group Linear correlation (1) vector Group's linear correlation vector group's rank _ linear algebra

Linear correlation of the first section vector group

A Mathematical Concepts

Defines 1.1 n ordered numbers, the array of which is called an n-dimensional vector, which is called n components of the vector, and the number I is called the first component.

Definition 1. 2 to the directional measure group A:, for any set of real numbers, vectors

A linear combination called a vector group A, called the coefficient of this linear combination.

Define 3 to the directional Group A: and vector β, if there is a set of numbers, so that

，

The vector β is a linear combination of vector Group A, at which point the vector β can be expressed linearly by the vector Group A.

Define 4 to the directional measure group A: If there is a group of not all zeros, so that

，

It is said that the vector Group A is linearly related, otherwise it is called linearly independent.

Definition 5 has two vector Group A:, and B: if each vector in Group B can be represented by a linear vector Group A, it is said that the B energy vector Group A is linear, if the vector group A and the vector Group B can be expressed linearly with each other, then the two vector groups are called equivalent. Two Principles, formulas, and rules

1. The basic principle of determining the linear correlation of a vector group:

When the upper form is set up, not all is 0, then can determine the linear correlation, if only, then can determine linearly independent.

2. The determination of the linear correlation of vectors

1) A vector A is a sufficient and necessary condition for linear correlation: a=0;

2 The sufficient and necessary conditions for the linear correlation of two vectors are: their corresponding components are proportional.

3 The sufficient and necessary conditions for the linear correlation of n-dimensional vectors are: the n-order determinant composed of them is zero.

4 The sufficient and necessary condition for the linear correlation of a vector group is that at least one vector in a vector group can be expressed linearly by the rest of the m-1 vectors.

5 The sufficient and necessary condition for the linear correlation of a vector group is that the rank of the matrix formed by it is less than the number m of the vector.

6 If the vector group is linearly correlated, then the vector group is also linearly correlated.

7 when M>n, M n-dimensional vectors must be linearly correlated.

8 The sufficient and necessary condition for a linear independence of vector A is: a≠0.

9 The sufficient and necessary conditions for the two vectors to be linearly independent are: their corresponding components are disproportionate.

The sufficient and necessary condition for the linear independence of n-dimensional vectors is that the n-order determinant composed of them is not equal to zero.

11 The sufficient and necessary condition for the linear independence of a vector group is that the rank of the matrix formed by it is equal to the number m of the vector.

12 the whole set of vectors are linearly independent, then any part of them is linearly independent.

13 If the vector group of R dimension is not linear, and after each vector in R dimension is added a component, then the vector of the r+1 dimension is also linearly independent.

3. If the vector group is linearly independent and the β is linearly correlated, the beta can be expressed linearly and the notation is unique.

4. The method of determining the linear correlation of the vector group: ① definition method, ② disprove method, ③ judgment method, ④ calculation method. Three Key points and difficulties analysis

The definition of this section, theorem, nature, inference more, and very abstract, difficult to understand, there is a certain degree of difficulty.

The emphasis is on the definition and understanding of the linear similarity of the vector group, and how to judge the linear correlation of a set of vectors.

Four Typical examples

Example 1. Set a vector group. When T is linearly correlated, the value of T is linearly independent.

Solution: Set

Obviously, when T=5, R (A) =2<3, so linear correlation.

When T 5 o'clock, R (A) = 3, so linear Independent.

Because the number of vectors is the same as the dimension of vectors, the linear correlation of the vector group can be determined by whether the 3-order determinant is equal to zero, which can be defined by the method of calculation.

Example 2. Vector β is represented linearly by a vector group, then the notation is the only sufficient necessary condition for line-line independent.

Proof: Necessity, set to make

(1)

and beta can be represented by linearity,

So there are (2)

Will (2)-(1) be

By the uniqueness of the notation, know:

Got, and so linearly irrelevant.

Sufficiency, suppose there are two representations, namely

Two-type subtraction.

Because of linear independence, so

So the notation is unique.

Section II. Rank one of the vector group. Digital Concepts

The definition 2.1 has a vector group A, and if there is an R vector in a, it satisfies

(1) Vector Group A0: linear Independent;

(2) any r+1 vector in a vector group A (if there is a r+1 vector in a) is linearly correlated, then the vector group A0 is a maximal linearly independent group of the Vector Group A (called the Maximal uncorrelated group).

The number of vectors in the maximum independent group of the 2.2 vector group is called the rank of the vector group.

The rank of the matrix column vector group is called the column rank of the matrix, and the rank in the row vector group is called the Matrix row rank. Two Principle, formula and law

Theorem 2.1 R (A) =a row rank =a column rank

Theorem 2.2 Vector Group A is equivalent to its maximal independent group.

Theorem 2.3 Set a vector Group B can be represented by a linear vector Group A, then the rank of the vector Group B is the rank of the vector Group A.

The inference 1 is equal to the rank of the equivalence vector group.

Inference 2 set, then.

Inference 3 Set the vector Group B is a part of the vector Group A, if the vector Group B is linearly independent, and the vector Group A can be expressed linearly by the vector Group B, then the vector Group B is a maximal independent group of the Vector Group A. Three Key points and difficulties analysis

The focus of this section is the definition of the maximal independent group and rank of a vector group, and the square of the maximal independent group and rank of the vector group

method, which is very important for the basic solution of the vector space and the homogeneous linear system. The difficulty is the proof of the theorem above, which requires the students to have some abstract thinking ability and logical ability. Four Typical examples

Example 1. Set a vector group

Calculate the rank of a vector group and a maximum independent vector group.

Analysis: This kind of problem can be solved according to the relation of matrix and vector Group and the rank of matrix column (row), and then the vector group can be spelled into a matrix, so that the rank of the highest order non 0 subtype of matrix A is the largest independent group of the Vector group.

Solution: Set, for a to perform elementary line transformation, get

Obviously r (A) = 2, so the rank of the vector group is 2, and is a maximum unrelated group.

Example 2. A vector group must be a linear combination of one of the largest unrelated groups in a vector group.

ID: Set Vector Group A: one of the largest independent group is A0:

Set ΑI is the number of vector Group A, divided into the following:

①αi in the vector group A0, there are

② if αi is not in the vector group A0, because A0 is the largest independent group of a and linearly correlated, αi can be expressed linearly.

Note the following two points for proof of the largest unrelated group:

(1) proving that the vector group is linearly independent;

(2) The total number of the whole vector group can be expressed linearly by the vector group, or the whole vector group is equivalent by using the equivalence relation.

From:http://dec3.jlu.edu.cn/webcourse/t000022/teach/index.htm