Lesson 4: Solution Sets -- Unique, Infinite, and None
Estimated time: 35-45 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Classify a system as consistent or inconsistent from its RREF
- Distinguish unique solutions from infinitely many solutions
- Identify free variables and express solutions in parametric form
- Write solution sets in parametric vector form
Three Possibilities for Solution Sets
A system of linear equations has exactly one of three outcomes:
Existence and Uniqueness:
- Exactly one solution (consistent, independent) -- the system has a unique answer.
- Infinitely many solutions (consistent, dependent) -- free variables parameterize a family of solutions.
- No solution (inconsistent) -- the equations contradict each other.
How to Tell from RREF
- No solution: A row of the form [0 0 ... 0 | b] with b nonzero (contradiction).
- Unique solution: No contradiction rows, and every column (except the last) is a pivot column.
- Infinitely many solutions: No contradiction rows, and at least one column (other than the last) is not a pivot column (free variable exists).
Case 1: Unique Solution
Example
RREF:
[ 0 1 0 | -1 ]
[ 0 0 1 | 4 ]
Every variable column is a pivot column. There are no free variables.
Solution: x_1 = 2, x_2 = -1, x_3 = 4. The solution set is the single point {(2, -1, 4)}.
Case 2: Infinitely Many Solutions
Example
RREF:
[ 0 1 -2 0 | 1 ]
[ 0 0 0 1 | -3 ]
[ 0 0 0 0 | 0 ]
Pivot columns: 1, 2, 4. Free variable: x_3 (column 3).
Let x_3 = t where t is any real number. Then:
- x_1 = 5 - 3t
- x_2 = 1 + 2t
- x_3 = t
- x_4 = -3
Parametric Vector Form
We can write the solution more elegantly by separating the constant part from the part that depends on the free variable:
Parametric Vector Form: Express the solution as a vector sum: a particular solution plus a linear combination of direction vectors scaled by free parameters.
Example: Writing in Parametric Vector Form
From the example above:
where t can be any real number. Here:
- (5, 1, 0, -3) is a particular solution (set t = 0)
- (-3, 2, 1, 0) is the direction vector corresponding to the free variable x_3
Geometrically, the solution set is a line through (5, 1, 0, -3) in the direction (-3, 2, 1, 0) in R^4.
Example: Two Free Variables
RREF:
[ 0 0 1 -1 | 2 ]
Pivot columns: 1, 3. Free variables: x_2 = s, x_4 = t.
From row 1: x_1 = 3 - 2s - t
From row 2: x_3 = 2 + t
The solution set is a plane in R^4, parameterized by two free variables.
Case 3: No Solution
Example
RREF:
[ 0 1 -1 | 3 ]
[ 0 0 0 | 1 ]
Row 3 says 0 = 1, which is impossible. The system is inconsistent. The solution set is empty.
Geometrically, the planes represented by the equations do not all share a common point.
Homogeneous Systems
Homogeneous System: A system where every constant on the right-hand side is zero: Ax = 0. A homogeneous system always has at least the trivial solution x = 0. It is always consistent.
The interesting question for homogeneous systems is: does it have nontrivial solutions (solutions other than x = 0)?
Key Fact
If a homogeneous system has more unknowns than equations (more columns than rows in the coefficient matrix), it must have a free variable and therefore infinitely many solutions.
Example: Homogeneous System
Solve Ax = 0 where the augmented matrix reduces to:
[ 0 1 -3 | 0 ]
Free variable: x_3 = t. Solution: x = t(- 2, 3, 1).
The solution set is a line through the origin. For homogeneous systems, the particular solution is always the zero vector, so the solution set is spanned by the direction vectors alone.
Check Your Understanding
1. A system in 4 unknowns reduces to RREF with 3 pivot columns. How many free variables are there? Is the solution unique?
2. Write the solution in parametric vector form: RREF is [ 1 -1 | 4 ] / [ 0 0 | 0 ].
3. A homogeneous system has 3 equations and 5 unknowns. Can the only solution be the trivial solution?
4. Classify the system whose RREF is [ 1 0 0 | 3 ] / [ 0 1 0 | -2 ] / [ 0 0 1 | 7 ].
5. How many direction vectors appear in the parametric vector form if there are 2 free variables?
Key Takeaways
- A system has no solution if RREF contains a row [0 ... 0 | b] with b nonzero
- A system has a unique solution if it is consistent and every variable column is a pivot column
- A system has infinitely many solutions if it is consistent and has at least one free variable
- Parametric vector form: x = p + t_1 v_1 + t_2 v_2 + ... where p is a particular solution and v_i are direction vectors
- Homogeneous systems (Ax = 0) are always consistent and have nontrivial solutions when there are more unknowns than equations
- Number of free variables = number of unknowns - number of pivots