Math 340 (row space, column space, null space of a matrix)
Let
A
= [
a
ij
] be an
m
by
n
matrix. Then
1. the
row space
of
A
is the subspace of
R
n
spanned by the rows of
A
, so all vectors of
the form
y
T
A
where
y
is in
R
m
;
2. the
column space
of
A
is the subspace of
R
m
spanned by the columns of
A
, so all
vectors of the form
Ax
where
x
is in
R
n
;
3. the
null space
of
A
is the subspace of
R
n
consisting of the solutions of the homogeneous
system
Ax
= 0.
Doing EROs is the same as multiplying on the left by an invertible (i.e., nonsingular
matrix)
E
: the matrix
EA
is obtained from
A
by doing EROs on
A
.
The eFect of EROs on an
m
by
n
matrix
A
, that is, the eFect on
A
by multiplying
A
on
the left by an invertible matrix, is thus:
1. ERO’s don’t change the row space of
A
, because
y
T
A
=
y
T
E

1
EA
= (
y
T
E

1
)(
EA
) =
z
T
(
EA
) where
z
T
=
y
T
E

1
,
but they can change the linear relations among the rows (so that rows that were linearly
independent become dependent after EROs and viceversa).
2. ERO’s don’t change the linear relations among the columns of
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Meyer
 Linear Algebra, Algebra, Vectors, Space, row space, ERO

Click to edit the document details