gaussian elimination

Complete

Summary

Linear systems as augmented matrices

Existence and Uniqueness Theorem

  • A linear system is consistent iff the rightmost column is not a pivot column

Number of parameters in general solution

for the matrix in RREF

Concept

Elementary row operations(reversible)

  • Interchanging,
  • Scaling,
  • Add multiple,

Gaussian elimination, reduce to REF

  1. Begin with leftmost nonzero column, it is a pivot column
  2. Interchange the top row if necessary, to bring a non-zero entry to the top of the column
  3. Add a multiple of the top row to the other rows to make the rest of the column 0
  4. Repeat 1-3 on the next row below
    Gauss-Jordan elimination, reduce to RREF
  5. Scale each row such that their leading entry is a 1
  6. Working back upwards, add multiples of lower rows to introduce 0s above the leading entries

Row equivalent matrices

they have the same solution set

Unknown constants in system

  • Consider cases where the pivot entries(in REF) with unknowns = 0

Intuition for row reduction

  • an expression of columns as multiples of the first n columns

Application

Row reduction

Consistent system (one solution)

Consistent systems (infinite solutions), insert a parameter

Inconsistent systems

Reduction with unknown constants

Extra

Octave

octave
# RREF of a matrix
rref(A)