📂Matrix Algebra

Simultaneous Homogeneous Linear Equations

Definition¹

In a linear system, if the constant terms are all $0$, it is called homogeneous.

$$\begin{align*} a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} &= 0 \\ a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} &= 0 \\ &\vdots \\ a_{m1}x_{1} + a_{m2}x_{2} + \cdots + a_{mn}x_{n} &= 0 \end{align*}$$

Unlike general linear systems, every homogeneous linear system always has a solution because if the constant terms are $0$, it obviously has $x_{1}=0, x_{2}=0, \dots, x_{n}=0$ as a solution. This is called the trivial solution. Solutions that are not trivial are called nontrivial solutions. Since a homogeneous linear system always has a trivial solution, there are only the following two cases for its solutions:

• Only the trivial solution exists.

• Both the trivial solution and infinitely many nontrivial solutions exist.

Theorem on Free Variables in Homogeneous Systems

Let’s consider a homogeneous linear system with $n$ unknowns. Suppose that the number of non-$0$ rows in the reduced row echelon form of the augmented matrix is $r$. Then, the homogeneous system can be represented simply as follows.

$$\begin{align*} & x_{1} & & & &+ (\quad) &= 0 & \\ & & x_{2} & & &+ (\quad) &= 0 & \\ & & & \ddots & & & \vdots & \\ & & & & x_{r} &+ (\quad) &= 0 & \end{align*}$$

This equation has $n-r$ free variables. Therefore, if $r < n$, there is at least one free variable, leading to infinitely many solutions. Hence, a homogeneous linear system with more unknowns than equations has infinitely many solutions.