MATH1850U/2050U:
Chapter 4 cont.
..
1
GENERAL VECTOR SPACES cont.
..
Row Space, Column Space, and Null Space (4.7; pg. 225)
Definition:
For an
n
m
matrix
mn
m
m
n
n
a
a
a
a
a
a
a
a
a
A
2
1
2
22
21
1
12
11
, the vectors in
R
n
formed from the
rows of
A
are called the
row vectors
of
A
, and the vectors in
R
m
formed from the columns of
A
are called the
column vectors
of
A
.
Example:
Identify the row vectors and column vectors of the matrix below.
Definition:
If
A
is an
n
m
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
0
x
A
,
which is a subspace of
R
n
, is called the
nullspace
of
A
.
Theorem:
A system of linear equations
b
x
A
is consistent if and only if
b
is in the column
space of
A
.
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 Spring '11
 PaulaTu
 Linear Algebra, Vectors, Vector Space, single solution, column vectors

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