Linear Algebra

128 Chapter 2. Vector Spaces
Another way, besides the prior result, to state that Gauss’ method has something
to say about the column space as well as about the row space is to consider
again Gauss-Jordan reduction. Recall that it ends with the reduced echelon form
of a matrix, as here.
⎛
⎞
⎛ ⎞
1
⎝2 3
6
1
3
6
16⎠−→
··· −→
1 3 1 6
1
⎝0 3
0
0
1
2
4⎠
0 0 0 0
Consider the row space and the column space of this result. Our first point
made above says that a basis for the row space is easy to get: simply collect
together all of the rows with leading entries. However, because this is a reduced
echelon form matrix, a basis for the column space is just as easy: take the
columns containing the leading entries, that is, 〈�e1,�e2〉. (Linear independence
is obvious. The other columns are in the span of this set, since they all have
a third component of zero.) Thus, for a reduced echelon form matrix, bases
for the row and column spaces can be found in essentially the same way —
by taking the parts of the matrix, the rows or columns, containing the leading
entries.
3.11 Theorem The row rank and column rank of a matrix are equal.
Proof. First bring the matrix to reduced echelon form. At that point, the
row rank equals the number of leading entries since each equals the number
of nonzero rows. Also at that point, the number of leading entries equals the
column rank because the set of columns containing leading entries consists of
some of the �ei’s from a standard basis, and that set is linearly independent and
spans the set of columns. Hence, in the reduced echelon form matrix, the row
rank equals the column rank, because each equals the number of leading entries.
But Lemma 3.3 and Lemma 3.10 show that the row rank and column rank
are not changed by using row operations to get to reduced echelon form. Thus
the row rank and the column rank of the original matrix are also equal. QED
3.12 Definition The rank of a matrix is its row rank or column rank.
So our second point in this subsection is that the column space and row
space of a matrix have the same dimension. Our third and final point is that
the concepts that we’ve seen arising naturally in the study of vector spaces are
exactly the ones that we have studied with linear systems.
3.13 Theorem For linear systems with n unknowns and with matrix of coefficients
A, the statements
(1) the rank of A is r
(2) the space of solutions of the associated homogeneous system has dimension
n − r
are equivalent.