Types of Solution Sets

We will now be more careful about analyzing the reduced row-echelon form derived from the augmented matrix of a system of linear equations. In particular, we will see how to systematically handle the situation when we have infinitely many solutions to a system, and we will prove that every system of linear equations has either zero, one or infinitely many solutions. With these tools, we will be able to routinely solve any linear system.

1. easily
2. Suppose we have converted the augmented matrix of a system of equations into reduced row-echelon form. How do we then identify the dependent and independent (free) variables?
3. What are the possible solution sets for a system of linear equations?

For Exercises C21C28, find the solution set of the system of linear equations. Give the values of $n$ and $r$, and interpret your answers in light of the theorems of this section.

For Exercises M51M57 say as much as possible about each system's solution set. Be sure to make it clear which theorems you are using to reach your conclusions.

The computer scientist Donald Knuth said, Science is what we understand well enough to explain to a computer. Art is everything else. In this section we will remove solving systems of equations from the realm of art, and into the realm of science. We begin with a definition.

We will want to first recognize when a system is inconsistent or consistent, and in the case of consistent systems we will be able to further refine the types of solutions possible. We will do this by analyzing the reduced row-echelon form of a matrix, using the value of $r$, and the sets of column indices, $D$ and $F$, first defined back in .

Use of the notation for the elements of $D$ and $F$ can be a bit confusing, since we have subscripted variables that are in turn equal to integers used to index the matrix. However, many questions about matrices and systems of equations can be answered once we know $r$, $D$ and $F$. The choice of the letters $D$ and $F$ refer to our upcoming definition of dependent and free variables (). An example will help us begin to get comfortable with this aspect of reduced row-echelon form.

The number $r$ is the single most important piece of information we can get from the reduced row-echelon form of a matrix. It is defined as the number of nonzero rows, but since each nonzero row has a leading 1, it is also the number of leading 1's present. For each leading 1, we have a pivot column, so $r$ is also the number of pivot columns. Repeating ourselves, $r$ is the number of nonzero rows, the number of leading 1's and the number of pivot columns. Across different situations, each of these interpretations of the meaning of $r$ will be useful, though it may be most helpful to think in terms of pivot columns.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

The next theorem simply states a conclusion from the final paragraph of the previous proof, allowing us to state explicitly the number of free variables for a consistent system.

We have accomplished a lot so far, but our main goal has been the following theorem, which is now very simple to prove. The proof is so simple that we ought to call it a corollary, but the result is important enough that it deserves to be called a theorem. (See .) Notice that this theorem was presaged first by and further foreshadowed by other examples.

Here is a diagram that consolidates several of our theorems from this section, and which is of practical use when you analyze systems of equations. Note this presumes we have the reduced row-echelon form of the augmented matrix of the system to analyze. Decision Tree for Solving Linear Systems

We have one more theorem to round out our set of tools for determining solution sets to systems of linear equations.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.