SYSTEMS of LINEAR EQUATIONS

Background: Applications
• Leontief Models in Economics
• Electric Circuits

Systems of Linear Equations Solution Types
• None - inconsistent system
• Only one (unique) - consistent system
• Infinitely many - consistent system

Matrix Notation Solution Method:
Use elementary row operations to simplify. Examples

ROW REDUCTION and ECHELON FORMS
Echelon Forms: nonzero entry, * = any number Examples
Uniqueness: Pivots:
• Pivot Position corresponds to a leading nonzero entry in a row of U
• Pivot Column contains a pivot position
Row Reduction Algorithm:
1.
2.
Interchange rows to find nonzero pivot
3.
Use row replacement ops to create zeros below pivot
4.
Repeat steps 1, 2, 3 on submatrix below current pivot row
Solution of Linear Systems:
• Basic variables correspond to pivot columns
• Free variables correspond to non pivot columns
• Backsubstitution determines xn first, then .
Existence and Uniqueness:
Consistent systems
• If and only if rightmost column of augmented U is not a pivot column
• Could have a unique solution (no free x's) or solutions

Alan C Genz
2000-07-06