Homogeneous System

Definition

A homogeneous system is a set of linear equations in which the coefficients of the variables and the constants on the right-hand side are all proportional. This means that the equations can be written in the form $Ax = 0$, where $A$ is a matrix of coefficients and $x$ is a vector of variables.

5 Must Know Facts For Your Next Test

1. The homogeneous system $Ax = 0$ has at least one solution, the zero vector $x = 0$, which is called the trivial solution.
2. The rank of the coefficient matrix $A$ determines the number of linearly independent solutions to the homogeneous system.
3. If the rank of $A$ is less than the number of variables, the homogeneous system has infinitely many non-trivial solutions.
4. The determinant of the coefficient matrix $A$ is zero for a homogeneous system, indicating that the system has either infinitely many solutions or no solution other than the trivial solution.
5. Solving a homogeneous system using determinants involves finding the values of the variables that make the determinant of the coefficient matrix equal to zero.

Review Questions

• Explain the relationship between the rank of the coefficient matrix and the number of solutions to a homogeneous system.
  • The rank of the coefficient matrix $A$ in a homogeneous system $Ax = 0$ determines the number of linearly independent solutions to the system. If the rank of $A$ is less than the number of variables, the system has infinitely many non-trivial solutions. This is because the null space of $A$ has a dimension greater than zero, meaning there are multiple linearly independent vectors $x$ that satisfy the equation $Ax = 0$. Conversely, if the rank of $A$ is equal to the number of variables, the system has only the trivial solution $x = 0$.
• Describe how the determinant of the coefficient matrix is used to solve a homogeneous system of equations.
  • The determinant of the coefficient matrix $A$ in a homogeneous system $Ax = 0$ plays a crucial role in determining the solution to the system. If the determinant of $A$ is zero, it indicates that the system has either infinitely many solutions or no solution other than the trivial solution $x = 0$. To solve the homogeneous system using determinants, one needs to find the values of the variables that make the determinant of $A$ equal to zero. These values represent the non-trivial solutions to the system, as the trivial solution $x = 0$ always satisfies the equation $Ax = 0$.
• Analyze the relationship between the homogeneous system $Ax = 0$ and the augmented matrix used to solve the system.
  • The augmented matrix, formed by combining the coefficient matrix $A$ and the constant terms (which are all zero in a homogeneous system), is a crucial tool for solving systems of linear equations, including homogeneous systems. By performing row reduction or Gaussian elimination on the augmented matrix, one can transform the system into an equivalent form that is easier to solve. The rank of the augmented matrix, which is the same as the rank of the coefficient matrix $A$, determines the number of linearly independent solutions to the homogeneous system. Understanding the connection between the homogeneous system $Ax = 0$ and the augmented matrix is essential for effectively solving these types of systems using determinants and other algebraic methods.

Related terms

Augmented Matrix:

The augmented matrix is a matrix formed by combining the coefficient matrix and the constant terms of a system of linear equations. It is used to solve the system using methods like Gaussian elimination or row reduction.

Determinant:

The determinant of a square matrix is a scalar value that is a function of the entries of the matrix. It is used to determine whether a system of linear equations has a unique solution, infinitely many solutions, or no solution.

Gaussian Elimination:

Gaussian elimination is a method for solving systems of linear equations by transforming the augmented matrix into an upper triangular form, which can then be used to find the values of the variables.