A Homogeneous System of Linear Equations

Homogeneous System

Definition 1. A system $Ax = b$ is said to be homogeneous if $b = 0$. Otherwise it is said to be nonhomogeneous.

    번역하면 homogeneous는 '동차', 즉 차수가 같다는 말이다. $b$는 방정식에서 상수항에 해당되고, 그 외의 항들은 모두 차수가 1인 미지수들이 곱해져 있다. 즉 $b$에 해당하는 항들을 제외하면 모두 미지수의 차수가 같으므로, 만일 상수항이 0이라면 상수항에 동일한 차수의 미지수를 곱한 것으로 생각할 수 있으므로 시스템 자체를 차수가 같은, 즉 동차 연립방정식이라고 볼 수 있다.

Theorem 1

Theorem 1. Let $Ax = \mathbf{0}$ be a homogeneous system, and let $K$ denote the solution set of $Ax = \mathbf{0}$. Then $K = N(L_A)$.

Proof. Note that the given system can be represented as $L_A(x) = Ax = \mathbf{0}$. Thus $K$ is exactly same to $N(L_A)$. $\blacksquare$

Corollary 1

Corollary 1. Any homogeneous system has at least one solution, namely, the zero vector.

    자명하게 $\mathbf{0} \in N(L_A)$이므로, 선형 동차 연립방정식은 적어도 하나의 해를 가지게 된다.

Corollary 2

Corollary 2. Let $Ax = \mathbf{0}$ be a homogenous system of $m$ linear equations in $n$ unknowns. If $m < n$, $Ax = \mathbf{0}$ has a nonzero solution.

Proof. Suppose that $Ax = \mathbf{0}$ has only zero soluton, i.e., $K = \{ \mathbf{0} \}$. Then dim($K$) = 0 = n - rank($A$) by The Dimension Theorem. $\Longrightarrow$ rank($A$) = $n$. Since $n$ is the maximum number of linearly independent rows of $A$, $n \leq m$. But $m < n \bigotimes$. Thus $Ax = \mathbf{0}$ has a nonzero solution. $\blacksquare$