Overview
This lecture introduces solving systems of linear equations using augmented matrices and row operations, explains legal operations, and demonstrates their application to different solution scenarios.
Row Operations: Legal Moves
- Three legal row operations: replacement, interchange, and scaling.
- Replacement: Replace a row by the sum of itself and a multiple of another row.
- Interchange: Swap two rows using appropriate notation.
- Scaling: Multiply a row by a nonzero constant (but never divide).
Solving with Augmented Matrices
- Convert system of equations to an augmented matrix before operations.
- Aim to create leading ones (pivot positions) and zeros below these positions.
- Use replacement to eliminate non-zero entries below pivots.
- Use scaling to turn pivot positions into ones.
- Use interchange if zero is in a required pivot position.
- Triangular form: matrix with zeros below the main diagonal (Gaussian elimination).
- For complete solution (Gauss-Jordan), create zeros above pivots as well.
Back Substitution
- After achieving triangular form, write each row as an equation.
- Start from the last row and substitute back to solve for all variables.
- Example: Find bottom variable first, substitute upward to solve remaining variables.
Consistency and Uniqueness
- Consistent system: At least one solution exists (previous examples).
- Inconsistent system: No solution exists (e.g., row with all zeros except last entry).
- Unique solution: Only one solution exists; otherwise, there may be infinite or none.
Practice Example Steps
- Always write system as augmented matrix first.
- Use row operations to form zeros under pivots and scale for leading ones.
- Apply multiple operations in steps if beneficial (e.g., dividing rows, combining row reductions).
- Keep reducing until solution emerges; express as ordered triples or pairs when finished.
Key Terms & Definitions
- Augmented Matrix — a matrix combining the coefficients and constants from a system of equations.
- Row Operations — legal manipulations (replacement, interchange, scaling) performed on matrix rows.
- Triangular Form — matrix form with zeros below main diagonal (“stairs”).
- Gaussian Elimination — process to reach triangular form via row operations.
- Gauss-Jordan Elimination — extends Gaussian elimination to zeros above pivots.
- Back Substitution — process of solving for variables starting from the bottom row upwards.
- Consistent System — system with at least one solution.
- Inconsistent System — system with no solution.
Action Items / Next Steps
- Practice converting systems to augmented matrices and using row operations.
- Attempt the provided practice question and check the solution step-by-step.
- Prepare for learning a formalized algorithm for systematic row reduction in next sessions.