Homogeneous and non-homogeneous systems of linear equations

Homogeneous system

 

Definition

Ax = 0 is called homogeneous.

 

Note

A homogeneous system always has the trivial solution x = 0.

 

Thm

If a homogeneous system has a non-zero solution, then it has $\infty$ solutions.

 

proof. Let $x_0$ be a solution. Then $\alpha x_0$ is also a solution. $\square$

 

Summary

 

In general, we have no / unique / $\infty$ solutions for sys of lin eqs.

 

For a homogeneous system, we have unique(x=0) / $\infty$ solutions.

 

Note

1. Ax=b has $\infty$ solutions $\Rightarrow$ so does Ax=0

2. Ax=b has a unique solution $\Rightarrow$ so does Ax=0

 

참고

Ax=b has no solution $\not\Rightarrow$ Ax = 0 has $\infty$ solution.

 

rank(A) $\leq$ n

rank(A|b) $\leq$ n+1

 

rank(A) = n < n+1 = rank(A|b) 인 경우, Ax = 0 은 unique solution 가진다.

rank(A) < n 인 경우, Ax = 0 은 $\infty$ solution 가진다.

 

Thm

If $x_0$ is a specific solution of Ax=b, then

 

Sol(Ax=b) = $x_0$ + Sol(Ax=0)

 

여기서 Sol(equation) 은 solution set of the equation을 의미하는 나만의 기호.

Sol(equation) = $\{x | equation\}$

 

e.g.

(x, x-3, 5-x, 1) : general solution of Ax=b / (0, -3, 5, 1) : specific solution of Ax=b

(x, x-3, 5-x, 1) = (0, -3, 5, 1) + (x, x, -x, 0)

$\therefore$ (x, x, -x, 0) : general solution of Ax=0

 

proof.

Ax=b, A$x_0$ = b $\Rightarrow$ A(x-$x_0$) = 0.

Ax=0, A$x_0$ = b $\Rightarrow$ A(x+$x_0$) = b. $\square$

 

Note

Ax = b가 해를 가질 때, Ax = b와 Ax = 0의 해집합이 같으니까 해의 개수도 당연히 같다.

 

결론

Thm 만 기억하면 된다!