Question

If a consistent (possibly nonhomogeneous) system of linear equations has more variables than equations, prove that it has more than one solution.

Short answer

The system has infinitely many solutions due to having at least one free variable.

Step-by-step solution

Identify the System

We have a consistent system of linear equations with more variables than equations. Let there be \( n \) variables and \( m \) equations where \( n > m \). This indicates an underdetermined system.

Understand Matrix Representation

Write the system in matrix form \( A\mathbf{x} = \mathbf{b} \), where \( A \) is an \( m \times n \) matrix, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the vector of constants. Here, \( A \) has more columns than rows since \( n > m \).

Explore Solution Space Dimensionality

According to the basic theorem of linear algebra, if a system is consistent, the solution set usually forms an affine space. The number of free variables is given by the difference in the number of variables and the rank of the matrix \( A \).

Determine the Rank and Free Variables

Since \( n > m \), the rank of \( A \) is at most \( m \). Let the rank of \( A \) be \( r \). The number of free variables, which is the solution space dimension, is \( n - r \). Since \( r \leq m < n \), \( n - r > 0 \), indicating the presence of at least one free variable.

Conclude Infinite Solutions

Having at least one free variable (\( n - r > 0 \)) implies that the system has a solution space of dimension greater than zero, indicating infinitely many solutions because the system can be parametrized by at least one parameter.

Key concepts

Systems of Linear Equations

A system of linear equations consists of multiple equations with multiple variables. The aim is often to find values for these variables that satisfy all equations simultaneously. In our specific problem, the system is underdetermined, which happens when there are more variables than equations. This can make solving the system more challenging but also interesting because it affects the number of solutions possible.
Key points to remember in systems of linear equations include:

• Each equation in the system represents a constraint on the variables.
• The goal is to search for solutions that can satisfy all constraints at the same time.
• An underdetermined system may lead to either no solutions or infinitely many solutions, depending on its consistency.

In our problem, we are dealing with a consistent system. Consistency means there is at least one solution, and due to more variables than equations, multiple or infinite solutions are possible.

Matrix Representation

The matrix representation of a system of linear equations is a compact and visual way of organizing the data involved. This is done by converting the system into the form \( A\mathbf{x} = \mathbf{b} \).
Here’s what each component represents:

• \( A \) is a matrix containing the coefficients of the variables from the equations, specifically an \( m \times n \) matrix when there are \( m \) equations and \( n \) variables.
• \( \mathbf{x} \) is a vector that represents the column of variables, which we aim to solve for.
• \( \mathbf{b} \) is a vector containing the constants on the right side of the equal signs in the equations.

In our case, since \( n > m \), the matrix \( A \) has more columns than rows. This highlights the fact that the system is underdetermined and suggests that the matrix will not have full row rank, playing a significant role in defining the size of the solution space.

Solution Space

The solution space in linear algebra refers to the set of all possible solutions that satisfy a system of equations. For consistent underdetermined systems, this space is not just a single point but a more complex set, often an affine space.
Understanding the dimensionality of the solution space is key:

• The rank of the matrix \( A \) reveals the number of dimensions spanned by \( A \).
• The dimension of the solution space is given by \( n - r \), where \( r \) is the rank of \( A \) and \( n \) is the number of variables.
• If \( n - r > 0 \), it means there exists at least one free variable, which allows for multiple solutions.

In our scenario, because there are more variables than the rank can accommodate, the solution space will have infinite dimensions. This results in an infinite number of solutions, each of which can be expressed in terms of free parameters. This outcome is integral to understanding how underdetermined systems behave and why they often produce a vast set of solutions.