Lecture 16: Systems of Nonhomogeneous Linear Equations

16.1 Systems of Nonhomogeneous Linear Equations

16.1.1 Basic Concepts

#####Definition: Nonhomogeneouslinearequations

description: $\begin{cases}a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n}=b_{1},\\a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n}=b_{2},\\\cdots\cdots\\a_{m1}x_{1}+a_{m2}x_{2}+\cdots+a_{mn}x_{n}=b_{m}\end{cases}$

Explanation

• Concept:
• It is called m equations and n systems of nonhomogeneous linear equations with unknown values;
• Therefore, we also get an extra column, which is the augmented matrix;
• Explanation:
• Basically, it's turning $x_{1}\alpha_{1}+x_{2}\alpha_{2}+\cdots+x_{n}\alpha_{n}=0$ into $x_{1}\alpha_{1}+x_{2}\alpha_{2}+\cdots+x_{n}\alpha_{n}=b$

16.1.2 Systems of Nonhomogeneous Linear Equations

Concept: Conditions for solutions

• If $r(A)\neq r([A,b])$, equivalently $\boldsymbol{b}$ cannot be represented as a linear combination of the columns $\alpha_1,\alpha_2,\cdots,\alpha_n$, then system (II) has no solution;
• -> Adding the column vector b increases the rank, so b is not in the column space of A;
• -> Therefore, Ax=b is inconsistent;
• If $r(A)=r([A,b])=n$, the columns $a_1,a_2,\cdots,a_n$ are linearly independent while $a_1,a_2,\cdots,a_n,b$ are linearly dependent, and system (II) has a unique solution;
• -> The rank equals the number of unknowns, so the coefficient matrix has full column rank;
• -> No free variables remain, so the solution is unique;
• If $r\left(A\right)=r\left(\left[A,b\right]\right)=r<n\quad$, then system (II) has infinitely many solutions.
• -> Degrees of freedom > the number of true constraints, so there are infinitely many solutions;

Concept: The property of solutions

• Let $\eta_1,\eta_2,\eta$ be solutions of the nonhomogeneous system $Ax=b$, and let $\xi$ be a solution of the corresponding homogeneous system $Ax=0$;
• (1) $\eta_1-\eta_2$ is a solution of Ax=0 -> the difference of any two solutions of the nonhomogeneous system is a solution of the homogeneous system;
• (2) $k\xi+\eta$ is a solution of Ax=b for any scalar $k$;

16.2 Solution Methods and Steps

Method: Method for solving nonhomogeneous linear equations

• Step 1:
• Write the derived system Ax=0 of Ax=b and find the general solution of Ax=0: $k_{1}\xi_{1}+k_{2}\xi_{2}+\cdots+k_{n-r}\xi_{n-r}$
• That is: first, as with homogeneous linear equations, find its general solution;
• Step 2:
• Find a particular solution $\eta$ for Ax=b
• That is, find one particular solution of the current nonhomogeneous system;
• Note: This particular solution is not unique;
• Step 3:
• The general solution of $Ax=b$ is $k_{1}\xi_{1}+k_{2}\xi_{2}+\cdots+k_{n-r}\xi_{n-r}+\eta$, where $k_{1},k_{2},\cdots,k_{n-r}$ are arbitrary constants;
• general solution of a nonhomogeneous system = general solution of the corresponding homogeneous system + one particular solution of the nonhomogeneous system;

Supplement: Conditions for solutions of nonhomogeneous systems -> Matrix rank

• Premise: Matrix $A_{m*n}$
• Full row rank: m=r(A)
• If $A$ has full row rank, its column space is all of $\mathbb{R}^m$, so $Ax=b$ is consistent for every $b\in\mathbb{R}^m$;
• Listed as full rank: n=r(A)
• If it is a homogeneous equation: the number of unknown variables equals the number of proper constraints;
• But in nonhomogeneous degrees, since it is r(A)=r(A|b)=n, it is not necessarily so;
• The condition for a system of equations to have infinitely many solutions: r(A)=r(A|b)<n

Conclusion: Regarding the solution to $A^{T}Ax=A^{T}b$

• If $A^{T}x=b$ has no solution -> $A^{T}Ax=A^{T}b$ find the best approximate solution;

Conclusion: $r(A)=r(A^T)=r(AA^T)=r(A^TA)$