Linear algebra: Chapter Three elementary transformations of matrices and linear equations (2) linear algebra of linear equations of Jie Yin

Section III Solution of a linear equation Group A. Mathematical concepts

According to the multiplication of matrices, the linear equations can be written in matrix form.

1. N-ary homogeneous linear equation group;

2. N-ary homogeneous linear equation group;

3. The coefficient matrix called A is the equation group, and b= (A,B) is the augmented matrix of the nonhomogeneous linear equation group. Two Principles, formulas, and rules

Theorem 3.1 The rank of a coefficient matrix A with sufficient and necessary conditions for a n-ary homogeneous linear equation set with a non 0 solution

R (A) <n.

Theorem 3.2 The rank of the coefficient matrix A with the sufficient and necessary conditions for the solution of N-ary systems of linear equations is equal to the rank of the augmented matrix b= (a,b).

It is obvious that theorem 3.1 is the problem of judging the solution of homogeneous linear equations, and theorem 3.2 is used to determine whether there is any solution to the nonhomogeneous linear equation group. Three. Analysis of key points and difficulties

This section focuses on the use of theorems 3.1, 3.2 to determine how homogeneous linear systems have solutions and nonhomogeneous linear equations. How to find out the solution of the equations and how to understand the proofs of Theorem 3.1 and 3.2 deeply. The proof of the theorem is simple and clear, but it is unique and difficult to grasp and understand with the many knowledge previously learned. Four. Typical examples

Example 1 solving homogeneous linear equation Group

Solution: The elementary row transformation to the coefficient matrix A is the simplest form matrix:

A system of equations with the same solution as the equation set

This is

Order, write it as a normal parameter form

Which is any real number, or written as a vector form

Example 2. A linear equation group is provided

When asked what value, this equation group (1) has a unique solution, (2) No solution, (3) has infinitely many solutions. And the general solution is obtained when there are infinitely many solutions.

Solution: the augmented matrix b= (A,B) is transformed into a row-step matrix by the elementary row transformation, which has

At that time, R (A) = r (B) = 3, the equation Group has a unique solution;

At that time, R (A) =1, R (B) = 2, the equation Group has no solution;

At that time, R (A) = r (B) = 2, the equation set has infinitely many solutions,

Was

And then we get the general solution.

That

From the above example, we know that for homogeneous linear equations, only the coefficient matrix must be converted into the simplest form matrix, and we can find the linear equations which are equivalent to the original Equation group, and then write the general solution. For nonhomogeneous linear equations, it is necessary to expand the matrix into a row-and-column ladder matrix, then we can judge whether it has a solution according to Theorem 3.2 and, when there is a solution, further convert the augmented matrix into the simplest form matrix to write out its general solution.

Section Fourth Elementary Phalanx I. Mathematical concepts

Definition 4.1 A matrix obtained by the unit matrix E through an elementary transformation is called a primary square.

1. Swap the two lines of the Unit matrix (column), get e[i,j];

2. By multiplying a row or column, get E[i (k)];

3. Multiply k by a line (column) and add to another row (column) up, get E[i,j (k)].

4. The elementary phalanx is reversible, and its inverse is still the same elementary phalanx. Two. Principle Formula and law

Theorem 4.1 Set A is a matrix for a primary row transformation, which is equal to the left of a is multiplied by the corresponding m-order elementary matrix, a primary column transformation for a, which is equivalent to a right multiplied by the corresponding n-order elementary Square.

Theorem 4.2 sets A to a reversible matrix, there is a finite elementary matrix.

The sufficient and necessary conditions for inference matrix a~b are: The existence of M-order invertible matrices P and N-order invertible matrix Q-Paq=b.

To find the inverse formula

Formula to be asked

Three. Analysis of key points and difficulties

The focus of this section is to find the inverse matrix of invertible matrices using elementary transformations. The difficulty is the method and technique of using elementary transformation to find the inverse matrix of invertible matrix, which is deduced from the formula above. Four. Typical examples

Example 1 set

。

Solution:

。

The method of using elementary row transformation to find the reversible matrix can also be used to find the matrix. By

If you are right (a| b To perform elementary row transformations, and B becomes when you turn A to E.

Example 2. Find matrix X to make ax=b, where

Solution: If a is reversible, then.

So

。

This example uses the Elementary row transformation method to obtain, if request, then the matrix can make the Elementary column transformation, causes

，

Can be. However, it is customary to make elementary row transformations, which can be changed to make elementary row transformations, so

，

Can be obtained, thus obtaining Y.

Section Fourth Elementary Phalanx I. Mathematical concepts

Definition 4.1 A matrix obtained by the unit matrix E through an elementary transformation is called a primary square.

1. Swap the two lines of the Unit matrix (column), get e[i,j];

2. By multiplying a row or column, get E[i (k)];

3. Multiply k by a line (column) and add to another row (column) up, get E[i,j (k)].

4. The elementary phalanx is reversible, and its inverse is still the same elementary phalanx. Two. Principle Formula and law

Theorem 4.1 Set A is a matrix for a primary row transformation, which is equal to the left of a is multiplied by the corresponding m-order elementary matrix, a primary column transformation for a, which is equivalent to a right multiplied by the corresponding n-order elementary Square.

Theorem 4.2 sets A to a reversible matrix, there is a finite elementary matrix.

The sufficient and necessary conditions for inference matrix a~b are: The existence of M-order invertible matrices P and N-order invertible matrix Q-Paq=b.

To find the inverse formula

Formula to be asked

Three. Analysis of key points and difficulties

The focus of this section is to find the inverse matrix of invertible matrices using elementary transformations. The difficulty is the method and technique of using elementary transformation to find the inverse matrix of invertible matrix, which is deduced from the formula above. Four. Typical examples

Example 1 set

。

Solution:

。

The method of using elementary row transformation to find the reversible matrix can also be used to find the matrix. By

If you are right (a| b To perform elementary row transformations, and B becomes when you turn A to E.

Example 2. Find matrix X to make ax=b, where

Solution: If a is reversible, then.

So

。

This example uses the Elementary row transformation method to obtain, if request, then the matrix can make the Elementary column transformation, causes

，

Can be. However, it is customary to make elementary row transformations, which can be changed to make elementary row transformations, so

，

Can be obtained, thus obtaining Y.