Definition
The n-ary linear equations whose constant terms are all 0
are called n-ary homogeneous linear equations. Suppose the coefficient matrix is A and the unknown item is X, then the matrix form is AX=0. If the number of non-zero rows of the row ladder-shaped matrix transformed by the elementary row transformation of its coefficient matrix is r, the solution of its system of equations has only the following two types:
When r=n When, the original equations have only zero solutions;
When r
Prove
The coefficient matrix of the aligned sublinear equations is transformed into a ladder type by elementary row transformation After the matrix, the number of rows r (that is, the rank of the matrix) that is not all zero is less than or equal to m (the number of rows of the matrix). Value, so that the original system of equations has non-zero solutions (infinitely many solutions).
Example
According to the theorem n=4>m=3, there must be a non-zero solution.
Perform elementary row transformation on the coefficient matrix:
The last matrix is the simplest form, and the homogeneous linear equation system of this coefficient matrix is:
Let X4 Are free variables, X1, X2, and X3 are the first variables.
Let X4=t, where t is any real number, and the solution of the original homogeneous linear equations is.
Judgment Theorem
Theorem 1
The necessary and sufficient condition for homogeneous linear equations to have non-zero solutions is r(A)
Corollary
The necessary and sufficient condition for homogeneous linear equations to have only zero solutions is r(A)=n.
Structure
The properties of the solution of homogeneous linear equations
Theorem 2 If x is a solution of homogeneous linear equations, then kx is also its solution , Where k is any constant.
Theorem 3 If x1, x2 are two solutions of homogeneous linear equations, then x1+x2 is also its solution.
Theorem 4 Align sublinear equations, if r(A)=r
there is a basic solution system, and the number of vectors contained in the basic solution system is nr, which is the dimension of the solution space The number is nr.Solution steps
1. Perform elementary row transformation on the coefficient matrix A and turn it into a row ladder matrix;
2, if r(A)=r=n (The number of unknowns), the original equations have only zero solutions, that is, x=0, the solution ends;
If r(A)=r
3, continue to The coefficient matrix A is transformed into the simplest row matrix, and the equations with the same solution are written out;
4. Select the appropriate free unknowns, and take the corresponding basic vector sets, and substitute them into the equations with the same solution to obtain The basic solution system of the original system of equations, and then write the general solution.
Properties
1. The sum of the two solutions of the homogeneous linear equations is still one of the homogeneous linear equations Assemble.
2. The k times of the solution of the homogeneous linear equations is still the solution of the homogeneous linear equations.
3. The rank of the coefficient matrix of the homogeneous linear equation system is r(A)=n, and the equation system has a unique zero solution.
The rank r(A) of the coefficient matrix of a homogeneous linear equation system
4. The necessary and sufficient condition for an n-ary homogeneous linear equation system to have a non-zero solution is that its coefficient determinant is Zero. Equivalently, the necessary and sufficient condition for the system of equations to have a unique zero solution is that the coefficient matrix is not zero. (Clem's Law)