[PubMed Mathematics] Linear Algebra Chapter 4 - Linear Equations (2, General Solutions of Linear Equations | Theoretical Extension)

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introduction

Continuing from the previous article, continue to learn the content of linear equations, starting from the general solution of the equations.

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4. General solutions of linear equations

4.1 Homogeneous linear equations

(1) Basic solution system - Let $r(A)=r<n$ , then$AX=0$ The maximum linearly independent group of the solution vector group formed by all solutions is called the equation system$AX=$A basic solution system of$0\; .$The number of linearly independent solution vectors contained in the basic solution system is$(n-r)$ pieces.

Because $r\left(A\right)=What\; about\; n$ ? because if$r\left(A\right)=If\; it\; is\; n$ , then the homogeneous equation has only zero solutions, and there is nothing to discuss.

When finding the basic solution system of a homogeneous linear equation system, the coefficient matrix is ​​transformed into steps by elementary row transformation (elementary row transformation of the coefficient matrix is ​​equivalent to the same solution transformation of the equation system), and the first non-zero element of each row is located The unknowns corresponding to the columns are constrained variables, and the remaining variables are free variables, so that the basic solution system can be determined (it is best to turn the first non-zero element of each row to 1, and turn the rest of the elements in the column to zero).

For example, suppose the equation system $AX=0$ coefficient matrix$After\; elementary\; row\; transformation,\; A$ can be reduced to the following form:

then$r(A)=3<5$ , the equation system$AX=$The basic solution system of$0$$n-r=5-3=2$ solution vectors, where$x_1,x_2,x_3$is the constraint variable, $x_4,x_5$is a free variable, $(x_4,x_5)$ take$(1,0)$ and$(0,1)$ , then the basic solution system is:$$X_1=(-2,1,-3,1,0{)}^T,X_2=(3,-4,2,0,1{)}^T.$$ (2) General solution——Suppose$X_1,X_2,\dots ,X_{n-r}$For the homogeneous linear equation system $AX=0$ , called$k_1{X}_1+k_2{X}_2+\cdots +k_{n-r}{X}_{n-r}$For the homogeneous linear equation system $AX=0$ , where$k_1,k_2,\dots ,k_{n-r}$for any constant.

4.2 Inhomogeneous linear equations

设 $r(A)=r(\overline{A})<n$，且$X_1,X_2,\dots ,X_{n-r}$for $AX=b$ 's derived equation system$AX=$A basic solution system of$0$${the}_0$for $AX=$A solution of$b$$AX=$The general solution of$b$$$k_1{X}_1+k_2{X}_2+\cdots +k_{n-r}{X}_{n-r}+{the}_0,$$ where$k_1,k_2,\dots ,k_{n-r}$for any constant.

1. Homogeneous linear equation system $AX=The\; basic\; solution\; system\; of\; 0$ is not unique, but the number of linearly independent solution vectors is unique.
2,$r(A)=r(\overline{A})<When\; n$ , the non-homogeneous linear equation system$AX=b$ The number of maximum linearly independent vectors of all solution vectors is$(n-r+1)$亪。
3，任${the}_1,{the}_2,\dots ,{the}_{n-r+1}$For non-homogeneous linear equations $AX=A\; maximal\; linear\; independent\; group\; of\; b$ , its general solution can also be expressed as$k_1{the}_1+k_2{the}_2+\cdots +k_{n-r+1}{the}_{n-r+1}$, where $k_1,k_2,\dots ,k_{n-r+1}$is any constant, and $k_1+k_2+\cdots +k_{n-r+1}=1.$

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5. Theoretical extension of the solution of equations

Theorem 1 - Let $A$ 是 $m\times n-$ matrix,$B$ is$n\times s$ matrix, if$AB=O$ , then$The\; column\; vector\; system\; of\; B$ is the equation system$AX=0$ solution.
Proof:Let$B=(b_1,b_2,\dots ,b_s)$，则$AB=(A{\beta }_1,A{\beta }_2,\dots ,A{\beta }_s)$，若$AB=O$，则$A{\beta }_1=0,A{\beta }_2=0\dots ,A{\beta }_s=0$ , the original proposition is proved.

Theorem 2 - Let the equation system $AX=0$ and$BX=0$ is the same solution equation system, then$r(A)=r\left(B\right)$ , and vice versa.

Theorem 3 - Let the equation system $AX=$The solution of$0$$BX=0Solution$ , for$r\left(A\right)\ge r(B).$

1. Let the equation system $AX=$The solution of$0$$BX=0$ , but not all, then$r\left(A\right)>r\left(B\right).$
2, let the equation system$AX=$The solution of$0$$BX=0$ , and$r(A)=r\left(B\right)$ , then the two equations have the same solution.

Theorem 4 —— Let $AX=b,BX=c$ , then the linear equation system$(A,B{)}^{T}X=(b,c{)}^{The\; solution\; of\; T}$ is the common solution of the two equations.