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Module 1: Quick Reference Card

Systems of Linear Equations -- Linear Algebra

Elementary Row Operations

Operations

  • Swap: R_i ↔ R_j
  • Scale: cR_i → R_i (c nonzero)
  • Replace: R_i + cR_j → R_i

Key Property

Row operations do NOT change the solution set. Row-equivalent matrices have the same solutions.

REF vs. RREF

REF (Row Echelon)

  • Zero rows at bottom
  • Pivots step right going down
  • Zeros below pivots

Solve by back-substitution.

RREF (Reduced)

  • All REF conditions PLUS
  • Every pivot = 1
  • Zeros above AND below pivots

Read solutions directly. Unique.

Solution Types

No solution: row [0 0 ... 0 | b], b nonzero

Unique Solution

Consistent + every variable column is a pivot column. No free variables.

Infinitely Many

Consistent + at least one free variable. Free vars = unknowns - pivots.

Parametric Vector Form

x = p + t_1 v_1 + t_2 v_2 + ...

p = particular solution (free vars = 0). v_i = direction vector (set i-th free var to 1, others to 0).

Homogeneous Systems (Ax = 0)

Always Consistent

x = 0 always works.

Nontrivial Solutions

Exist when #unknowns > #pivots (guaranteed if #unknowns > #equations).

Quick Algorithm: Gauss-Jordan

1. Forward eliminate (create zeros below pivots, left to right) → REF.
2. Scale pivots to 1.
3. Back-eliminate (create zeros above pivots, right to left) → RREF.
4. Read solution or write parametric form.