AcademyLinear Systems
Academy
Solution Sets
Level 1 - Linear Algebra topic page in Linear Systems.
Principle
The solution set of a linear system is the set of all vectors that satisfy every equation at the same time. A linear system can have no solution, exactly one solution, or infinitely many solutions.
Row reduction reveals which case occurs.
Notation
\(A\mathbf x=\mathbf b\)
a linear system
\(\mathbf x\)
a candidate solution vector
\(\text{pivot variable}\)
a variable corresponding to a pivot column
\(\text{free variable}\)
a variable corresponding to a non-pivot column
\(t,s\)
real parameters used for free variables
The Core Test
After reducing the augmented matrix:
- If there is a row \(0=\text{non-zero}\), there is no solution.
- If every variable column has a pivot and there is no inconsistent row, there is a unique solution.
- If at least one variable column has no pivot and there is no inconsistent row, there are infinitely many solutions.
The inconsistent row pattern is
Inconsistent row
\[\begin{pmatrix}0&0&\cdots&0&|&c\end{pmatrix},\qquad c\ne0\]
Free variables become parameters in the solution set.
Worked Cases
Question
Classify the solution set represented by \(\begin{pmatrix}1&0&|&2\\0&1&|&-3\end{pmatrix}\).
Answer
Both variable columns are pivot columns. The rows read \(x=2\) and \(y=-3\). There is no inconsistent row and no free variable. The system has exactly one solution: \((2,-3)\).
Examples
Question
Classify a system whose reduced augmented matrix contains \(\begin{pmatrix}0&0&|&7\end{pmatrix}\).
Answer
The row says \(0=7\), which is impossible. Therefore no vector satisfies all equations at the same time, and the solution set is empty.