Solution Sets of Linear Systems

In linear algebra, the solution set of a linear system encompasses all possible solutions to a given system of equations. The nature of the solution set depends on the system’s consistency and the rank of the coefficient matrix.

Types of Solution Sets

Unique Solution:
The system has exactly one solution if it is consistent and the coefficient matrix has full column rank. For a square matrix, this occurs when the determinant is nonzero.
Infinite Solutions:
The system has infinitely many solutions when it is consistent, but the coefficient matrix has less than full column rank. The solution set forms a subspace or affine space.
No Solution:
The system is inconsistent when the augmented matrix has a rank greater than the coefficient matrix, meaning the equations contradict each other.

Mathematical Representation

A linear system can be written in matrix form as:

$A\mathbf{x} = \mathbf{b},$

where:

$A$ is the $m \times n$ coefficient matrix,
$\mathbf{x}$ is the $n \times 1$ vector of variables,
$\mathbf{b}$ is the $m \times 1$ constants vector.

The solution set depends on the relationship between $A$ and $\mathbf{b}$ :

Unique Solution: A single vector $\mathbf{x}$ satisfies $A\mathbf{x} = \mathbf{b}$ :

$\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}.$

Infinite Solutions: The solution set is represented parametrically as:

$\mathbf{x} = \mathbf{p} + c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_k\mathbf{v}_k,$

where $\mathbf{p}$ is a particular solution, and $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ are the basis vectors of the null space of $A$ .

No Solution: The system has no representation since $A\mathbf{x} \neq \mathbf{b}$ for all $\mathbf{x}$ .

Example

Consider the system:

$\begin{aligned} x + y + z &= 6, \\ 2x + y - z &= 3, \\ x - y + 2z &= 4. \end{aligned}$

The augmented matrix is:

$\begin{bmatrix} 1 & 1 & 1 & 6 \\ 2 & 1 & -1 & 3 \\ 1 & -1 & 2 & 4 \end{bmatrix}.$

Using row reduction, the matrix becomes:

$\begin{bmatrix} 1 & 1 & 1 & 6 \\ 0 & -1 & -3 & -9 \\ 0 & 0 & 0 & 0 \end{bmatrix}.$

The solution set is:

$\mathbf{x} = \begin{bmatrix} 3 \\ -2 \\ 5 \end{bmatrix} + t\begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}, \quad t \in \mathbb{R}.$

Geometric Interpretation

Unique Solution: Represents a single point in $\mathbb{R}^n$ .
Infinite Solutions: Represents a line, plane, or higher-dimensional subspace in $\mathbb{R}^n$ .
No Solution: No intersection in $\mathbb{R}^n$ .

Key Concepts

Null Space: The set of solutions to the homogeneous equation $A\mathbf{x} = \mathbf{0}$ .
Particular Solution: A specific solution to the non-homogeneous equation $A\mathbf{x} = \mathbf{b}$ .
Consistency: Determined by comparing the rank of the coefficient matrix $A$ and the augmented matrix $[A \ \mathbf{b}]$ .

Applications

Solution sets of linear systems are widely used in:

Physics: Modeling equilibrium in forces or electric circuits.
Economics: Solving optimization problems with constraints.
Engineering: Designing structures or analyzing systems of equations.

Understanding solution sets is a foundational concept in linear algebra and provides a gateway to more advanced topics like eigenvalues, eigenvectors, and least squares solutions.

Solution Sets of Linear Systems