MATH 4377, ADVANCED LINEAR ALGEBRA, SUMMER 2010, Review for Ex3
Warning
:
This is only a partial review.
You still need to read the
textbook to see all the topics we have covered.
Read key to HWs and do
more practices using exercises on the textbook.
1.
How to use the Gauss elimination process to get the reduced
echelon form
?
Practice Problem
: Let
A
=
1

2

1
1
2

3
1
6
3

5
0
7
1
0
5
9
.
(a) Find the rowreduced echelon matrix
R
of
A
.
(b) Find the elementary matrices
E
1
, . . . , E
k
such that
E
k
· · ·
E
1
A
=
R.
2. Solving homogeneous system of linear equations
Ax
= 0, where
A
is a
m
×
n
matrix:
•
Its solution space is a
vector space
, which is the same as the nullspace
of
L
A
. The dimension of the solution space is
n

rank
(
A
).
•
Solutions
x
(complete solutions)of
Ax
= 0 can be written as a linear
combinations of the vectors in a basis of the solution space, i.e.
x
=
c
1
x
1
+
· · ·
+
c
k
x
k
, where
{
x
1
, . . . , x
k
}
is a basis for the solution space,
and
k
=
n

rank
(
A
).
•
Ax
= 0 is equivalent to
Rx
= 0 where
R
is the educed echelon form of
A
. Hence a basis for the solution space
{
x
1
, . . . , x
k
}
can be found by
looking at
R
.
•
Practice problem (see HW#9)
: Suppose
A
has row reduced form
R
,
A
=
1
2
1
b
2
a
1
8
?
?
?
?
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 Summer '08
 Staff
 Linear Algebra, Algebra, ax, solution space

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