1. Key Definitions
Linear Equation: a_1 x_1 + a_2 x_2 + ... + a_n x_n = b. Variables to the first power only, no products of variables.
System of Linear Equations: A collection of linear equations in the same variables. A solution must satisfy ALL equations.
Augmented Matrix: A matrix capturing coefficients and constants: [A | b].
Row Equivalent: Matrices related by elementary row operations. They represent systems with the same solution set.
2. Elementary Row Operations
| Operation | Notation | Description |
|---|
| Row Swap | R_i ↔ R_j | Interchange two rows |
| Row Scaling | cR_i → R_i (c nonzero) | Multiply a row by a nonzero constant |
| Row Replacement | R_i + cR_j → R_i | Add a multiple of one row to another |
3. Row Echelon Form (REF)
Conditions: (1) Zero rows at bottom. (2) Pivots step right going down. (3) Zeros below each pivot.
After reaching REF, solve by back-substitution (bottom row up).
4. Reduced Row Echelon Form (RREF)
REF conditions PLUS: (4) Every pivot = 1. (5) Each pivot is the only nonzero entry in its column.
RREF is unique for any matrix. Solutions can be read directly.
5. Solution Classification
| Type | RREF Indicator | Geometry |
|---|
| No solution | Row [0 0 ... 0 | b], b nonzero | Planes do not intersect |
| Unique solution | No contradictions, all variable columns are pivot columns | Planes meet at a single point |
| Infinitely many | No contradictions, at least one free variable | Planes meet along a line/plane |
6. Parametric Vector Form
x = p + t_1 v_1 + t_2 v_2 + ...
where p = particular solution (set free variables to 0), and v_i = direction vector for the i-th free variable.
7. Homogeneous Systems (Ax = 0)
Always consistent (x = 0 is always a solution). If #unknowns > #equations, nontrivial solutions exist.
Solution set of Ax = 0 is always x = t_1 v_1 + t_2 v_2 + ... (no particular solution needed since p = 0).
8. Key Facts to Remember
- Number of free variables = (number of unknowns) - (number of pivots)
- RREF is unique; REF is not unique
- Row operations preserve the solution set
- A consistent system with n unknowns and n pivots has a unique solution
- A consistent system with free variables has infinitely many solutions