In this section we present the rank theorem, which is the culmination of all of the work we have done so far.

The reader may have observed a relationship between the column space and the null space of a matrix. In this example in Section 3.3, the column space and the null space of a matrix are both lines, in and respectively:

In this example in Section 3.1, the null space of the matrix is a plane in and the column space the line in spanned by

In this example in Section 3.1, the null space of a matrix is a line in and the column space is a plane in

In all examples, the dimension of the column space plus the dimension of the null space is equal to the number of columns of the matrix. This is the content of the rank theorem.

The rank of a matrix gives us important information about the solutions to Recall from this note in Section 2.4 that is consistent exactly when is in the span of the columns of in other words when is in the column space of Thus, is the dimension of the set of with the property that is consistent.

We know that the rank of is equal to the number of pivot columns (see this theorem in Section 3.4), and the nullity of is equal to the number of free variables (see this theorem in Section 3.4), which is the number of columns without pivots. To summarize:

Clearly

so we have proved the following theorem.

In other words, for any consistent system of linear equations,

The rank theorem theorem is really the culmination of this chapter, as it gives a strong relationship between the null space of a matrix (the solution set of ) with the column space (the set of vectors making consistent), our two primary objects of interest. The more freedom we have in choosing the less freedom we have in choosing and vice versa.

The rank theorem is a prime example of how we use the theory of linear algebra to say something qualitative about a system of equations without ever solving it. This is, in essence, the power of the subject.