[Go] Solving Linear Equations

§ 4 solutions of Linear Equations

TheoremN-element Linear Equations

1. No Solution

2. There is a unique solution

3. Infinite Solutions

Proof: For the convenience of discussion, it is recommended that the augmented matrix be transformed into the simplest matrix of the following rows after several elementary row transformations

Certificate

1. Then, the R + 1 line of the above matrix corresponds to the contradictory equation, so the equations do not exist.

2. The simplest matrix of the above rows is

The corresponding equations are:

That is, the equations have a unique solution.

3. Then, the corresponding equations can be expressed

The solution of the equations containing parameters is obtained:

That is

Since the parameters are optional, the equations have an infinite number of solutions.

Example: Solving Homogeneous Linear Equations

Solution

Solving Linear Equations

Solution

Linear Equations for Example

When I ask what value, the equations have a unique solution, no solution, or an infinite solution? And obtain the general solution when there are infinite solutions.

Solution

1. The equations have a unique solution;

2. There is no solution to the equations;

3. The equations have infinite solutions and the general solution is

Some promotions:

1. matrix equations have solutions.

2., then.

3. matrix equations only have zero solutions.

[Go] Solving Linear Equations