Question

Explain what is meant by the trivial solution of a system of linear equations and what is meant by a non-trivial solution.

Short answer

Answer: In a system of linear equations, a trivial solution is one where all variables have a value of zero, whereas a non-trivial solution is a solution in which at least one variable has a non-zero value. The difference between the two lies in the values assigned to the variables, with the trivial solution being the "zero solution" and non-trivial solutions having at least one non-zero variable.

Step-by-step solution

Definition of Trivial Solution

A trivial solution is the solution to a system of linear equations where all variables are equal to zero. In other words, in a trivial solution, every variable in the given equations has a value of zero. This is also known as the "zero solution".

Example of Trivial Solution

Let's consider the following system of linear equations: \begin{cases} x + y + z = 0 \\ 2x + 3y - z = 0 \\ 3x - y + 2z = 0 \end{cases} The trivial solution to this system would be x = 0, y = 0, and z = 0, which satisfies all three equations.

Definition of Non-Trivial Solution

A non-trivial solution is a solution to a system of linear equations in which at least one variable has a non-zero value. In other words, in a non-trivial solution, not all variables have a value of zero, making the solution different from the trivial solution.

Example of Non-Trivial Solution

Let's consider the following system of linear equations: \begin{cases} x - y = 1 \\ x + y = 3 \end{cases} By solving these equations, we find that x = 2, and y = 1. Since both variables have non-zero values, the solution is a non-trivial solution.

Key concepts

System of Linear Equations

Understanding the basics of a system of linear equations is crucial for delving into more complex algebraic concepts. A system of linear equations consists of two or more equations that are solved simultaneously. Each equation represents a line on a graph, and the solution to the system is the point or points where the lines intersect.

Let’s demystify this concept with an example. Consider the system given by \begin{cases}3x + 2y = 6\x - y = 0\text{System A}\text{End of System A}\text{Here, we have two equations involving two variables:} \(x\) and \(y\). To find a solution, we are looking for values of \(x\) and \(y\) that make both equations true at the same time. If you can graph these equations, their intersection would reveal the solution.

Solving systems of linear equations can involve different methods, such as substitution, elimination, or graphical analysis. Sometimes, a system can have

• No solution (the lines are parallel and never intersect)
• Exactly one solution (the lines intersect at one point)
• Infinitely many solutions (the lines are coincidental and overlap entirely)

Comprehending how to solve these systems is fundamental because they appear in various fields of study, from physics to economics.

Zero Solution (Trivial Solution)

The zero solution, also known as the trivial solution, of a system of linear equations happens when all variables in the system are set to zero, resulting in each equation being satisfied. This often occurs in homogenous systems, where the constant term is zero.For instance, if we take the system \begin{cases} x + 3y - 5z = 0 \\ 2x - y + 4z = 0 \\ -3x + 7y + z = 0 \end{cases}, the trivial solution would be x=0, y=0, and z=0. This solution is termed 'trivial' because it is the most straightforward solution to identify and provides minimal information about the relationships between the variables.

Non-Zero Solution (Non-Trivial Solution)

In contrast to the trivial solution, a non-trivial or non-zero solution is where at least one of the variables in the system does not equal zero. These solutions are more informative as they offer insight into the relationship between variables within the system.

Let’s consider a simple system \begin{cases}2x + 4y = 8\\y - x = 1\end{cases}. A non-trivial solution to this system would be values of \(x\) and \(y\) that are not zero and simultaneously satisfy both equations. For example, possible solutions include x=2, y=2 or x=3, y=4, among others.

Such solutions are significant because they can represent multiple scenarios in real-life applications. For example, in the context of economics, non-trivial solutions can correspond to different pricing strategies that satisfy both demand and supply equations. In physics, they can correspond to different velocities that comply with equations of motion in a dynamic system.