A system of linear equations : A collection of one or more linear equations
Solution Set : The Set of all possible solutions of the linear system
→ Two linear systems are called equivalent if they have the same solution set.
A system of linear equations has either
- No Solution ( INCONSISTENT )
- Exactly one Solution ( CONSISTENT )
- Infinitely Many Solutions ( CONSISTENT )
Matrix Notation
- 3 rows x 3 columns => Coefficient Matrix ( 3 \(\times\) 3 ) : 계수 행렬
- 3 rows x 4 columns => Augmented Matrix ( 3 \(\times\) 4 ) : 확대 행렬
- 첨가 행렬은 계수 행렬과 상수항들의 행렬을 맞붙여 얻는 행렬
Augmented Matrix
\(A = \left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 7 & 4 \\ 1 & 6 & 8 \\ \end{matrix} \right] , B = \left[ \begin{matrix} 1 \\ 4 \\ 1 \\ \end{matrix} \right]\) 가 있을 때, Augmented Matrix 는 \(\left[ \begin{matrix} 1 & 2 & 3 & 1\\ 4 & 7 & 4 & 4 \\ 1 & 6 & 8 & 1 \\ \end{matrix} \right]\) 이다.
Elementary row operations
- Replacement
- Interchange
- Scaling
Two matrices are Row Equivalent If there is a sequence of elementary row operations that transforms one matrix into the other.
If the augmented matrices of the linear systems are row equivalent, then the two systems have the same solution set.