Nontrivial — Definition, Meaning & Examples

Nontrivial

A solution or example that is not trivial. Often, solutions or examples involving the number zero are considered trivial. Nonzero solutions or examples are considered nontrivial.

For example, the equation x + 5y = 0 has the trivial solution (0, 0). Nontrivial solutions include (5, –1) and (–2, 0.4).

Key Formula

a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = 0

Where:

$a_1, a_2, \ldots, a_n$ = Known coefficients of the equation
$x_1, x_2, \ldots, x_n$ = Unknown variables to solve for
$(0, 0, \ldots, 0)$ = The trivial solution — all variables equal zero
$\text{Any solution where at least one } x_i \neq 0$ = A nontrivial solution

Worked Example

Problem: Find a nontrivial solution to the equation 2x + 3y = 0.

Step 1: Identify the trivial solution. Setting both variables to zero gives (0, 0), which satisfies the equation.

2(0) + 3(0) = 0 \quad \checkmark

Step 2: To find a nontrivial solution, choose a nonzero value for one variable. Let y = 2.

2x + 3(2) = 0

Step 3: Solve for x.

2x + 6 = 0 \implies 2x = -6 \implies x = -3

Step 4: Verify the solution by substituting back into the original equation.

2(-3) + 3(2) = -6 + 6 = 0 \quad \checkmark

Answer: The nontrivial solution is (x, y) = (−3, 2). At least one variable is nonzero, so this solution is nontrivial.

Another Example

This example shows a homogeneous system with multiple equations and three variables, illustrating the key theorem that a homogeneous system with more unknowns than independent equations always has nontrivial solutions.

Problem: Determine whether the system x + y + z = 0 and 2x + 2y + 2z = 0 has a nontrivial solution.

Step 1: Notice the second equation is just twice the first. This means both equations give the same constraint: x + y + z = 0. The system effectively has one equation and three unknowns.

x + y + z = 0

Step 2: Because there are more unknowns than independent equations, infinitely many solutions exist. Choose y = 1 and z = 1.

x + 1 + 1 = 0 \implies x = -2

Step 3: Verify the nontrivial solution in both original equations.

(-2) + 1 + 1 = 0 \quad \checkmark \qquad 2(-2) + 2(1) + 2(1) = -4 + 2 + 2 = 0 \quad \checkmark

Answer: Yes. One nontrivial solution is (x, y, z) = (−2, 1, 1). In fact, infinitely many nontrivial solutions exist because the system has more unknowns than independent equations.

Frequently Asked Questions

What is the difference between trivial and nontrivial solutions?

A trivial solution is the obvious, zero-based solution that works for any homogeneous equation — typically all variables set to zero. A nontrivial solution is any solution where at least one variable is nonzero. For example, in the equation 3x − y = 0, the trivial solution is (0, 0), while (1, 3) is a nontrivial solution.

When does a homogeneous system always have a nontrivial solution?

A homogeneous system of linear equations always has a nontrivial solution when the number of unknowns exceeds the number of independent equations. This is guaranteed by a fundamental result in linear algebra. For instance, one equation in three unknowns always has infinitely many nontrivial solutions.

Can a nonhomogeneous equation have a trivial solution?

It depends. The term 'trivial' is most commonly applied to homogeneous equations (those set equal to zero), where the all-zeros solution is always available. For a nonhomogeneous equation like x + y = 5, the point (0, 0) does not satisfy the equation at all, so the concept of trivial vs. nontrivial is less relevant in that context.

Nontrivial vs. Trivial

	Nontrivial	Trivial
Definition	A solution where at least one variable is nonzero	The obvious, zero-based solution (all variables equal zero)
Typical example for ax + by = 0	Any (x, y) ≠ (0, 0) satisfying the equation	(0, 0)
Existence	Not always guaranteed; depends on the system	Always exists for homogeneous equations
Usefulness	Provides meaningful information about the system	Provides no new information — merely confirms the equation is valid

Why It Matters

You encounter the concept of nontrivial solutions throughout algebra and linear algebra, especially when working with homogeneous systems of equations. Determining whether a system has only the trivial solution or also nontrivial solutions tells you whether the system's equations are truly independent. In applications like physics and engineering, nontrivial solutions correspond to meaningful physical states — for example, the vibration modes of a bridge or the eigenvectors of a matrix.

Common Mistakes

Mistake: Assuming that any solution containing a zero component is trivial.

Correction: A solution is trivial only when ALL variables are zero. The solution (5, 0) to the equation x + 5y = 0 is nontrivial because x = 5 is nonzero, even though y = 0.

Mistake: Thinking nontrivial solutions always exist for every homogeneous system.

Correction: A homogeneous system can have only the trivial solution. For example, x + y = 0 and x − y = 0 has the unique solution (0, 0). Nontrivial solutions are guaranteed only when there are more unknowns than independent equations.

Related Terms

Trivial — The opposite concept — the obvious zero solution
Solution — Values satisfying an equation or system
Zero — The value that typically defines a trivial solution
Nonzero — A nontrivial solution has at least one nonzero variable
Equation — The statement in which solutions are found
Homogeneous System of Equations — System set equal to zero; always has trivial solution
System of Equations — Multiple equations solved simultaneously