The number of solutions in a system of linear equations

Definition

rank(A) : # of nonzero rows in a ref(A)

 

A가 $n\times m$ 행렬이라고 할 때, REF의 모양을 생각해보면 rank(A) $\leq$ min(# of rows, # of columns)

 

Note
one can define column-operation, column echelon form, rank in terms of columns ...

하지만, 이렇게 정의하면 solution set이 바뀐다.

그럼에도 불구하고 the rank defined by row = the rank defined by columns.

 

The number of solutions in a system of linear equation

Let Ax = b be a system of m equations with n unknowns.

Then,

1. rank(A) = rank(A|b) = n $\Leftrightarrow$ The system has a unique solution.

2. rank(A) != rank(A|b) $\Leftrightarrow$ The system has no solution 

3. rank(A) = rank(A|b) < n $\Leftrightarrow$ The system has infinitely many solutions with n-rank(A) degrees of freedom.

Here, (A|b) : augmented matrix

 

Note

rank(A) $\neq$ rank(A|b) 라는 것은 ref를 구했을 때 0 = 3 같은 방정식을 얻게 된다는 것.

따라서, no solution.

 

rank(A) = rank(A|b) 라는 것은 consistent informations 만 있다는 뜻.

따라서, there is a solution.

이 때, rank(A) means # of consistent informations A can preserve.