Key Concepts of Systems of Linear Equations

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Why This Matters

Systems of linear equations form the backbone of linear algebra—they're the framework you'll use to analyze everything from circuit networks to economic models to computer graphics transformations. When you encounter these systems, you're really being tested on your ability to recognize solution structures, apply systematic elimination techniques, and interpret what the algebra tells you about the underlying geometry. The concepts here—consistency, rank, independence—will resurface constantly throughout the course.

Don't just memorize definitions and procedures. For each concept below, understand what it reveals about the system's behavior and when you'd choose one method over another. Exam questions love to ask you to identify solution types from matrix forms, explain why a particular method works, or connect algebraic results to geometric interpretations. Master the "why" behind each technique, and you'll handle anything the exam throws at you.

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Foundations: Setting Up the System

Before you can solve anything, you need to understand how systems are structured and represented. These foundational concepts establish the language we use to describe and manipulate linear systems.

Definition of a System of Linear Equations

• A collection of linear equations sharing the same variables—each equation represents a hyperplane in $n$-dimensional space
• The solution set consists of all variable values that satisfy every equation simultaneously—geometrically, this is where all hyperplanes intersect
• Variables appear only to the first power with no products between them—this linearity is what makes systematic solution methods possible

Coefficient Matrix and Augmented Matrix

• The coefficient matrix $A$ contains only the variable coefficients, arranged with each equation as a row—essential for analyzing system properties like rank
• The augmented matrix $[A|b]$ appends the constant terms after a vertical line—this is your working matrix for elimination methods
• Row operations on the augmented matrix preserve the solution set while transforming the system into easier-to-solve forms

Vector Form of Linear Systems

• Expresses the system as $A\mathbf{x} = \mathbf{b}$—a single matrix equation replacing multiple scalar equations
• Geometrically interprets solutions as finding the linear combination of column vectors of $A$ that produces $\mathbf{b}$
• Compact notation simplifies theoretical analysis and connects to broader concepts like column space and linear transformations

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Classifying Solutions: What Can Happen?

Every system falls into one of three categories. Understanding why each occurs—and how to identify which you're dealing with—is fundamental to everything that follows.

Consistent and Inconsistent Systems

• Consistent systems have at least one solution—the equations' geometric representations share at least one common point
• Inconsistent systems have no solutions—you'll see a row like $[0 \; 0 \; \cdots \; 0 \; | \; c]$ where $c \neq 0$ in row echelon form, representing a contradiction
• Consistency depends on the relationship between rank of $A$ and rank of $[A|b]$—if they're equal, the system is consistent

Unique, Infinite, and No Solutions

• Unique solution occurs when the system has exactly as many independent equations as unknowns—geometrically, $n$ hyperplanes intersecting at a single point
• Infinitely many solutions arise when equations are redundant (dependent), creating a line, plane, or higher-dimensional solution space
• No solution means the equations contradict each other—parallel hyperplanes that never meet

Homogeneous Systems

• All constant terms equal zero, written as $A\mathbf{x} = \mathbf{0}$—the zero vector is always a solution (trivial solution)
• Always consistent since $\mathbf{x} = \mathbf{0}$ works—the question is whether non-trivial solutions exist
• Non-trivial solutions exist when there are more variables than independent equations (free variables present)

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The Elimination Toolkit: Transforming Systems

These are your primary tools for systematically solving systems. Master the operations and the forms they produce.

Elementary Row Operations

• Three permitted operations: swap two rows, multiply a row by a non-zero scalar, add a scalar multiple of one row to another
• Solution-preserving transformations—the modified system has exactly the same solutions as the original
• Foundation of all matrix reduction methods—Gaussian elimination is just strategic application of these operations

Gaussian Elimination

• Systematic forward elimination transforms the augmented matrix into row echelon form using elementary row operations
• Creates a "staircase" pattern of leading entries (pivots), making the system triangular and solvable by back-substitution
• Reveals solution type during the process—contradictory rows indicate inconsistency, free variables indicate infinite solutions

Row Echelon Form and Reduced Row Echelon Form

• Row echelon form (REF) has all-zero rows at the bottom, each leading entry right of the one above, and zeros below each pivot
• Reduced row echelon form (RREF) additionally requires each pivot to be 1 and the only non-zero entry in its column
• RREF gives solutions directly without back-substitution—each pivot column's variable equals the corresponding constant

Back-Substitution

• Works backward from the last equation in row echelon form, substituting known values into earlier equations
• Required after Gaussian elimination to REF—you solve the bottom equation first, then work upward
• Unnecessary with RREF since the matrix already displays the solution explicitly

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Measuring System Structure: Independence and Rank

These concepts tell you how much information your equations actually contain—crucial for predicting solution behavior before you solve.

Linear Independence and Dependence

• Independent equations each contribute unique constraints—none can be derived from combinations of others
• Dependent equations contain redundancy—at least one equation is a linear combination of the rest, adding no new information
• Determines solution uniqueness: $n$ independent equations in $n$ unknowns yield a unique solution; dependence creates free variables

Rank of a Matrix

• The number of linearly independent rows (or columns)—equivalently, the number of pivots in row echelon form
• Rank-nullity theorem: $\text{rank}(A) + \text{nullity}(A) = n$ (number of columns), connecting rank to solution space dimension
• Solution classification: if $\text{rank}(A) = \text{rank}([A|b]) = n$, unique solution; if ranks are equal but less than $n$, infinite solutions; if ranks differ, no solution

Parametric Solutions

• Express infinite solution sets using free variables as parameters—each free variable can take any real value
• General solution form: particular solution plus any linear combination of homogeneous solutions
• Number of parameters equals $n - \text{rank}(A)$, the dimension of the solution space

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Alternative Solution Methods

Beyond Gaussian elimination, these methods offer different advantages depending on system size and structure.

Cramer's Rule

• Uses determinants to solve square systems ($n$ equations, $n$ unknowns) where the coefficient matrix is invertible
• Formula: $x_i = \frac{\det(A_i)}{\det(A)}$ where $A_i$ replaces column $i$ of $A$ with the constant vector
• Practical only for small systems ($2 \times 2$ or $3 \times 3$)—determinant computation becomes expensive for larger matrices

Matrix Inverse Method

• Solves $A\mathbf{x} = \mathbf{b}$ directly as $\mathbf{x} = A^{-1}\mathbf{b}$ when $A$ is square and invertible
• Requires $\det(A) \neq 0$—if the determinant is zero, $A$ has no inverse and this method fails
• Efficient when solving multiple systems with the same coefficient matrix but different constant vectors—compute $A^{-1}$ once, multiply repeatedly

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Quick Reference Table

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Self-Check Questions

1. Given an augmented matrix in row echelon form, how do you determine whether the system is consistent, and if consistent, whether the solution is unique or infinite?

2. Compare Gaussian elimination to the matrix inverse method: what conditions must be met to use each, and when would you prefer one over the other?

3. A homogeneous system $A\mathbf{x} = \mathbf{0}$ has 5 variables and $\text{rank}(A) = 3$. How many free parameters appear in the general solution, and why is the system guaranteed to have non-trivial solutions?

4. What's the relationship between the rank of the coefficient matrix and the rank of the augmented matrix for consistent systems? How does this change for inconsistent systems?

5. If you're asked to solve a $3 \times 3$ system for only one specific variable and you know the coefficient matrix is invertible, which method—Gaussian elimination, Cramer's rule, or matrix inverse—would be most efficient, and why?