Classification of Systems of Linear Equations

Linear Equations Nomenclature

• The rank of a system of linear equations is the number of linearly independent rows or columns of the coefficients matrix of that system. The coefficients matrix is a grid of the numbers that precede the system variables. In our example, the coefficients matrix would be:

  4 5

  4 -2

  For a row (or column) to be linearly independent of another row (or column), it must be the case that one row (or column) cannot be produced by a linear combination of another row (or column). You shouldn't be able to multiple all the elements of row 1 by a single number to get row 2. You can see that all the columns in our example coefficients matrix are linearly independent because there exists no single number that would allow us to multiply 4 to get 5 and -2. You can also see that the rows in our example matrix are linearly independent. There exists no single number that when multiplied by 4 produces 4, and when multiplied by 5 produces -2. This means the rank of our example system is 2.

  The augmented matrix is a combination of the coefficients matrix and the solution vector. In our example the augmented matrix would be:

  4 5 1

  4 -2 2

  Because this matrix has two rows, the highest value the rank of the augmented matrix can possibly be is 2. Therefore, for this example, the rank of the augmented matrix is equal to the rank of the matrix of coefficients.

Extending the System

• In our example system of equations, there are only two variables. The equations describe lines in two-dimensional space. If we were to add another set of variables the equations would describe planes in three-dimensional space. This can be extended to multiple dimensions. Instead of thinking in terms of systems with any particular number of variables, we can think in terms of a generic system with n variables. This enables us to classify the general properties of all systems of equations regardless of the number of variables in the system.

No Solution

• If the rank of the coefficients matrix is not equal to the rank of the augmented matrix, there is no solution. There is no unique set of values that fulfills the requirements described in the system of equations. The system of equations cannot be solved. If the system cannot be solved, the system is said to be inconsistent.

A Unique Solution

• There is a single, unique set of solutions to the system of equations if the rank of the coefficients matrix is equal to the rank of the augmented matrix and they are both equal to the number of columns of the coefficients matrix. There is a single set of values that fulfills the requirements described by the system of equations. If there is a unique solution, the system is said to be independent.

An Infinite Number of Solutions

• The system of equations has an infinite number of solutions if the rank of the coefficients matrix is equal to the rank of the augmented matrix and they are both less than the number of rows in the coefficients matrix. Thiere is an infinitely large set of values that fulfill the requirements described by the system of equations. If there are an infinite number of solutions, the system is said to be dependent.