# 1.5: Rank and Homogeneous Systems

- Page ID
- 14500

There is a special type of system which requires additional study. This type of system is called a homogeneous system of equations, which we defined above in Definition [def:homogeneoussystem]. Our focus in this section is to consider what types of solutions are possible for a homogeneous system of equations.

Consider the following definition.

If the system has a solution in which not all of the \(x_1, \cdots, x_n\) are equal to zero, then we call this solution **nontrivial**. The trivial solution does not tell us much about the system, as it says that \(0=0\)! Therefore, when working with homogeneous systems of equations, we want to know when the system has a nontrivial solution.

Suppose we have a homogeneous system of \(m\) equations, using \(n\) variables, and suppose that \(n > m\). In other words, there are more variables than equations. Then, it turns out that this system always has a nontrivial solution. Not only will the system have a nontrivial solution, but it also will have infinitely many solutions. It is also possible, but not required, to have a nontrivial solution if \(n=m\) and \(n<m\).

Consider the following example.

Suppose we were to write the solution to the previous example in another form. Specifically, \[\begin{array}{c} x = 0 \\ y = 0 + t \\ z = 0 + t \end{array}\] can be written as \[\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 0\\ 0\\ 0 \end{array} \right] + t \left[ \begin{array}{r} 0\\ 1\\ 1 \end{array} \right]\] Notice that we have constructed a column from the constants in the solution (all equal to \(0\)), as well as a column corresponding to the coefficients on \(t\) in each equation. While we will discuss this form of solution more in further chapters, for now consider the column of coefficients of the parameter \(t\). In this case, this is the column \(\left[ \begin{array}{r} 0\\ 1\\ 1 \end{array} \right]\).

There is a special name for this column, which is **basic solution**. The basic solutions of a system are columns constructed from the coefficients on parameters in the solution. We often denote basic solutions by \(X_1, X_2\) etc., depending on how many solutions occur. Therefore, Example [exa:homogeneoussolution] has the basic solution \(X_1 = \left[ \begin{array}{r} 0\\ 1\\ 1 \end{array} \right]\).

We explore this further in the following example.

We now present a new definition.

A remarkable result of this section is that a linear combination of the basic solutions is again a solution to the system. Even more remarkable is that every solution can be written as a linear combination of these solutions. Therefore, if we take a linear combination of the two solutions to Example [exa:basicsolutions], this would also be a solution. For example, we could take the following linear combination

\[3 \left[ \begin{array}{r} -4 \\ 1 \\ 0 \end{array} \right] + 2 \left[ \begin{array}{r} -3 \\ 0\\ 1 \end{array} \right] = \left[ \begin{array}{r} -18 \\ 3 \\ 2 \end{array} \right]\] You should take a moment to verify that \[\left[ \begin{array}{r} x \\ y \\ z \end{array} \right] = \left[ \begin{array}{r} -18 \\ 3 \\ 2 \end{array} \right]\]

is in fact a solution to the system in Example [exa:basicsolutions].

Another way in which we can find out more information about the solutions of a homogeneous system is to consider the **rank** of the associated coefficient matrix. We now define what is meant by the rank of a matrix.

Similarly, we could count the number of pivot positions (or pivot columns) to determine the rank of \(A\).

Notice that we would have achieved the same answer if we had found the row-echelon form of \(A\) instead of the reduced row-echelon form.

Suppose we have a homogeneous system of \(m\) equations in \(n\) variables, and suppose that \(n > m\). From our above discussion, we know that this system will have infinitely many solutions. If we consider the rank of the coefficient matrix of this system, we can find out even more about the solution. Note that we are looking at just the coefficient matrix, not the entire augmented matrix.

Consider our above Example [exa:basicsolutions] in the context of this theorem. The system in this example has \(m = 2\) equations in \(n = 3\) variables. First, because \(n>m\), we know that the system has a nontrivial solution, and therefore infinitely many solutions. This tells us that the solution will contain at least one parameter. The rank of the coefficient matrix can tell us even more about the solution! The rank of the coefficient matrix of the system is \(1\), as it has one leading entry in row-echelon form. Theorem [thm:rankhomogeneoussolutions] tells us that the solution will have \(n-r = 3-1 = 2\) parameters. You can check that this is true in the solution to Example [exa:basicsolutions].

Notice that if \(n=m\) or \(n<m\), it is possible to have either a unique solution (which will be the trivial solution) or infinitely many solutions.

We are not limited to homogeneous systems of equations here. The rank of a matrix can be used to learn about the solutions of any system of linear equations. In the previous section, we discussed that a system of equations can have no solution, a unique solution, or infinitely many solutions. Suppose the system is consistent, whether it is homogeneous or not. The following theorem tells us how we can use the rank to learn about the type of solution we have.