Chapter 4: Problem 58

Discussion Make up a system of three linear equations in three variables for which the solution set is \(\\{(0,0,0)\\} .\) A system with this solution set is called a homogeneous system. Why do you think it is given that name?

### Short Answer

## Step by step solution

## Understanding Homogeneous Systems

## Choosing Coefficients for the Equations

## Verify the Solution Set

## Explanation of the Term 'Homogeneous'

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### linear algebra

Homogeneous systems are a subset of linear equations where all constant terms are zero. This means every equation in the system takes the form: \[ a_1x + b_1y + c_1z = 0 \]\[ a_2x + b_2y + c_2z = 0 \]\[ a_3x + b_3y + c_3z = 0 \]

To solve such a system, we apply techniques from linear algebra like Gaussian elimination, matrix operations, and row reduction.

Understanding the foundations of linear algebra, such as vector spaces, linear transformations, and eigenvalues, is crucial. These concepts provide tools to solve various types of linear systems, including homogeneous systems.

- Vector spaces
- Linear transformations
- Eigenvalues and eigenvectors
- Matrix operations

###### solution set

In our specific system:\[ x + y + z = 0 \]\[ 2x - y + 3z = 0 \]\[ -x + 4y - z = 0 \]We need to verify that \( (0, 0, 0) \) satisfies all three equations, which we did by substituting the values and confirming that each equation balances.

For homogeneous systems of linear equations, if there are multiple solutions, they lie along a line, plane, or higher-dimensional analogue passing through the origin \( (0, 0, 0) \). This origin is called a 'trivial solution', and any non-zero solutions are 'non-trivial solutions'.

- Trivial solution: \( (0, 0, 0) \)
- Non-trivial solutions: Any solution that is not \( (0, 0, 0) \)

###### three variables

In a homogeneous system like ours:\[ x + y + z = 0 \]\[ 2x - y + 3z = 0 \]\[ -x + 4y - z = 0 \]We're working in a three-dimensional space. The solution set can be represented as a point, line, or plane within this space.

For three variables, we commonly utilize matrix representation and row reduction techniques to find solutions. The equations can be set up in a matrix form: \[ \begin{bmatrix} 1 & 1 & 1 \ 2 & -1 & 3 \ -1 & 4 & -1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 0 \ 0 \ 0 \ \end{bmatrix} \]

By applying linear algebra techniques, we can solve the system. The presence of three variables often requires more complex operations like row reduction to find the solution set.

- Dimension: Three-dimensional space
- Matrix representation
- Row reduction technique