Row Space and Column Space
References are to Anton–Rorres
PROBLEM: Compute everything about the 4×5 matrix
1 −2 0 0 3
2 −5 −3 −2 6
A=
(1)
0 5 15 10 0
2 6 18 8 6
(This includes Example 8 (p. 267) in §5.5.)
The nullspace of A Find the dimension (= nullity(A)) and a basis. In effect, solve
the linear system Ax = 0. Therefore we use elementary row operations to reduce A
to row echelon form (not uniquely, so your answer may vary)
1 −2 0 0 3
0 1 3 2 0
R= (2)
0 0 1 1 0
0 0 0 0 0
with the leading ones boxed. Or we can go all the way to reduced row echelon form
1 0 0 −2 3
0 −1 0
0
0 1
R = (3)
0 0 1 1 0
0 0 0 0 0
which is unique. From (2) or (3), it is clear that x4 and x5 are free variables and may
be chosen arbitrarily; we put x4 = r and x5 = s. Then from (2), by back substitution,
or directly from (3), the general solution of Ax = 0 can be written
x1 = 2r − 3s, x2 = r, x3 = −r, x4 = r, x5 = s
Thus the nullspace has dimension 2, as it needs two coordinates, and has the basis
{(2, 1, −1, 1, 0), (−3, 0, 0, 0, 1)}
(Here, the first vector is obtained by setting r = 1 and s = 0 and the second by r = 0
and s = 1; equivalently, we read off the coefficients of r and s in each xj .)
The row space of A Find the dimension (= rank(A)) and a basis. By Theorem
5.5.4, the row space of A is the same as the row space of R (or R0 ). But by Theorem
5.5.6, we see from (2) that the first three rows of R form a basis. (None of these
rows is a linear combination of later rows, and the zero row has no effect on the row
space.) Thus the row space of A has dimension rank(A) = 3 and has the basis
{(1, −2, 0, 0, 3), (0, 1, 3, 2, 0), (0, 0, 1, 1, 0)}
The column space of A Find the dimension (= rank(A)) and a basis. Write uj
for column j of R0 . It is clear that u1 = e1 , u2 = e2 , and u3 = e3 , and that these form
110.201 Linear Algebra JMB File: rowcol, Revision A; 27 Aug 2001; Page 1
2 Row Space and Column Space
a basis of the column space of R0 . Explicitly, we read off that u4 = −2u1 − u2 + u3
and u5 = 3u1 . The column space of R0 is not the same as the column space of A;
however, Theorem 5.5.5 allows us to conclude that the corresponding columns cj of
A do the same job for A. Namely, the column space of A has dimension rank(A) = 3
and has the basis
{c1 = (1, 2, 0, 2), c2 = (−2, −5, 5, 6), c3 = (0, −3, 15, 18)}
Further, the remaining columns of A are expressed in terms of these as
c4 = (0, −2, 10, 8) = −2c1 − c2 + c3 , c5 = (3, 6, 0, 6) = 3c1
as is easily checked from (1).
The row space of A, revisited Better: Select a basis from the rows of A,
{r1 , r2 , r3 , r4 }. Theorem 5.4.6 says that this is always possible, and sometimes, this
is what we need. We just did it for the columns of A. [It is perfectly possible to
develop a theory of column operations, but we and Anton–Rorres choose not to go
this route.] IDEA: Consider the transpose of A, the 5×4 matrix
1 2 0 2
−2 −5 5 6
AT =
0 −3 15 18
0 −2 10 8
3 6 0 6
Elementary row operations reduce this to the row echelon form (see p. 268)
1 2 0 2
0 1 −5 −10
0 0 0 1 (4)
0 0 0 0
0 0 0 0
and to the reduced row echelon form
1 0 10 0
0 1 −5 0
0 0 0 1
(5)
0 0 0 0
0 0 0 0
Write vj for column j of this matrix. This time, we have the basis {v1 , v2 , v4 }, and
v3 = 10v1 − 5v2 . Again by Theorem 5.5.5 (applied to AT ), we deduce the basis
{r1 = (1, −2, 0, 0, 3), r2 = (2, −5, −3, −2, 6), r4 = (2, 6, 18, 8, 6)}
of the row space of A, and the relation (easily verified from (1))
r3 = (0, 5, 15, 10, 0) = 10r1 − 5r2
The nullspace of AT Find the dimension and a basis. From (5), we see that this
time there is only one free variable, x3 . The dimension is 1 and the basis consists of
the single vector (−10, 5, 1, 0). Note that 1 = 4 − 3, as in Theorem 5.6.3 (for AT ).
110.201 Linear Algebra JMB File: rowcol, Revision A; 27 Aug 2001; Page 2
