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1
ROW SPACE, COLUMN SPACE, and NULLSPACE
EXAMPLE
List the Row and Column Vectors in 3
ä
4
Matrix
12
3
3
4
21
0
1
35
71
14
27
The row vectors are
(2
1 0 1),
(3 5 7
1),
(1 4 2 7)
Thecolumn vectors are
0
1
3,
5,
7,
1
2
7
Solution
A
−
⎛⎞
⎜⎟
=−
⎝⎠
=
−
=
−
⎛ ⎞
⎜ ⎟
==
=
=
−
⎝ ⎠
rr
r
cc
c
c
DEFINITION
If
A
is an
m
x
n
matrix, then the subspace of
R
n
spanned
by the row vectors of
A
is called the
row
space
of
A,
and the subspace of
R
m
spanned by the column vectors
of
A
is called the
column space
of A.
The solution space of the homogeneous system of
equations
A
x
=
0
, which is a subspace of
R
n
,
is called the
nullspace
of A.
_
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THEOREM 5.5.1
A system of linear equations
A
x
=
b
is consistent if
and only if
b
is in the column space of
A.
The next theorem establishes a fundamental
relationship between the solutions of a
nonhomogeneous linear system
A
x
=
b
and those of
the corresponding homogeneous linear system
A
x
=
0
with the same coefficient matrix.
THEOREM 5.5.2
If
x
0
denotes any single solution of a consistent linear
system
A
x
=
b
and
v
1,
v
2
,…,
v
k
form a basis for the
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 Winter '07
 KIRUSCHEVA

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