Linear Algebra Exam 1 Study
Guide (Sections 1.1–2.3)
This study guide summarizes the key concepts, theorems, and examples
from Lay's Linear Algebra and its Applications (5th Edition), covering
Sections 1.1 through 2.3. It focuses specifically on the concepts required for
the problems assigned by your professor.
Section 1.1 — Systems of Linear Equations
• A linear equation has the form a₁x₁ + a₂x₂ + ... + aₙxₙ = b.
• The solution set of a system is the set of all values (x₁, ..., xₙ) that satisfy
every equation.
• Systems can be:
- Consistent: at least one solution (unique or infinite).
- Inconsistent: no solution.
• Elementary Row Operations (EROs):
1. Replace a row by itself + multiple of another row.
2. Interchange two rows.
3. Multiply a row by a nonzero scalar.
• Augmented matrix: combines coefficients and constants.
• A system has:
- No solution → row [0 0 0 | 1]
- One solution → each variable is a pivot variable.
- Infinite solutions → one or more free variables.
Section 1.2 — Row Reduction and Echelon Forms
• Echelon Form Rules:
1. Nonzero rows above zero rows.
2. Each leading entry is to the right of the one above it.
3. Entries below each pivot are zero.
• Reduced Echelon Form adds:
4. Each pivot is 1.
5. Pivots are the only nonzero entries in their columns.
• Pivot columns → basic variables. Nonpivot columns → free variables.
• Free variables lead to infinitely many solutions.
