Solution Sets

Principle

A linear system can have exactly one solution, no solution, or infinitely many solutions. Row reduction shows which case applies by revealing pivots, free variables, and inconsistent rows.

For a matrix equation \(A\mathbf{x}=\mathbf{b}\), solutions are vectors \(\mathbf{x}\) that make the equation true.

Notation

• \(A\mathbf{x}=\mathbf{b}\) is an inhomogeneous system when \(\mathbf{b}\ne\mathbf{0}\).
• \(A\mathbf{x}=\mathbf{0}\) is the associated homogeneous system.
• The trivial solution of a homogeneous system is \(\mathbf{x}=\mathbf{0}\).
• A particular solution \(\mathbf{x}_p\) is one specific solution to \(A\mathbf{x}=\mathbf{b}\).
• A homogeneous solution \(\mathbf{x}_h\) is any solution to \(A\mathbf{x}=\mathbf{0}\).

Method

To describe the solution set of a linear system:

1. Write the augmented matrix for \(A\mathbf{x}=\mathbf{b}\).
2. Row reduce the augmented matrix.
3. Check for an inconsistent row such as \(0=1\).
4. Identify pivot variables and free variables.
5. If there is a unique solution, read off the variable values.
6. If there are free variables, assign parameters and write the solution set.

Rules

• An inconsistent row means the system has no solution.
• If every variable is a pivot variable and the system is consistent, the system has a unique solution.
• If at least one variable is free and the system is consistent, the system has infinitely many solutions.
• A homogeneous system \(A\mathbf{x}=\mathbf{0}\) always has the trivial solution \(\mathbf{x}=\mathbf{0}\), so it is always consistent.
• Free variables produce infinitely many solutions because their parameters can vary over infinitely many real values.
• If \(\mathbf{u}\) and \(\mathbf{v}\) both solve \(A\mathbf{x}=\mathbf{b}\), then \(\mathbf{u}-\mathbf{v}\) solves \(A\mathbf{x}=\mathbf{0}\).
• Every solution to a consistent inhomogeneous system has the form \(\mathbf{x}=\mathbf{x}_p+\mathbf{x}_h\), where \(\mathbf{x}_p\) solves \(A\mathbf{x}=\mathbf{b}\) and \(\mathbf{x}_h\) solves \(A\mathbf{x}=\mathbf{0}\).

Examples

Suppose row reduction of an augmented matrix gives

\[ \left(\begin{array}{ccc|c} 1&0&2&5\\ 0&1&-1&1\\ 0&0&0&0 \end{array}\right). \]

The variables \(x\) and \(y\) are pivot variables, while \(z\) is free. Let \(z=t\), where \(t\in\mathbb R\). The rows give

\[ x+2z=5, \qquad y-z=1. \]

Therefore

\[ \begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}5\\1\\0\end{pmatrix} +t\begin{pmatrix}-2\\1\\1\end{pmatrix}, \qquad t\in\mathbb R. \]

Here \(\mathbf{x}_p=\begin{pmatrix}5\\1\\0\end{pmatrix}\) is a particular solution and \(\mathbf{x}_h=t\begin{pmatrix}-2\\1\\1\end{pmatrix}\) is the homogeneous part.

If \(\mathbf{u}\) and \(\mathbf{v}\) both solve \(A\mathbf{x}=\mathbf{b}\), then

\[ A(\mathbf{u}-\mathbf{v})=A\mathbf{u}-A\mathbf{v}=\mathbf{b}-\mathbf{b}=\mathbf{0}, \]

so their difference is a homogeneous solution.

Checks

• Check for inconsistent rows before deciding whether free variables give infinitely many solutions.
• Check that an inconsistent row such as \(0=1\) means no solution, not one free variable.
• Check whether each variable is a pivot variable or a free variable.
• Check that a homogeneous system includes the trivial solution \(\mathbf{x}=\mathbf{0}\).
• Check that the particular solution \(\mathbf{x}_p\) solves \(A\mathbf{x}=\mathbf{b}\), while the homogeneous part \(\mathbf{x}_h\) solves \(A\mathbf{x}=\mathbf{0}\).