Math 1b Practical — Basic matrices and solutions; pivot operations
January 3, 2011
We use matrices to model systems of linear equations. For example, the system
2
x
1
−
3
x
2
−
5
x
3
−
7
x
4
+
2
x
5
=
7
x
1
+
x
2
−
5
x
3
+
x
4
−
4
x
5
=
3
x
1
−
4
x
3
+
3
x
4
+
x
5
=
−
4
(1)
has corresponding matrix
2
−
3
−
5
−
7
2
1
1
−
5
1
−
4
1
0
−
4
3
1
7
3
−
4
.
(2)
Two matrices (of the same dimensions) are
rowequivalent
when one can be obtained
from the other by a sequence of elementary row operations as described in LADW, Sec
tion 2.1 on page 40. It is important to understand that the systems of linear equations
corresponding to two row equivalent matrices will have exactly the same set of solutions.
An
r
by
n
matrix
M
with all nonzero rows is
basic
when its columns include the unit
vectors
⎛
⎜
⎜
⎜
⎜
⎝
1
0
0
.
.
.
0
⎞
⎟
⎟
⎟
⎟
⎠
,
⎛
⎜
⎜
⎜
⎜
⎝
0
1
0
.
.
.
0
⎞
⎟
⎟
⎟
⎟
⎠
,
⎛
⎜
⎜
⎜
⎜
⎝
0
0
1
.
.
.
0
⎞
⎟
⎟
⎟
⎟
⎠
,
. . . ,
⎛
⎜
⎜
⎜
⎜
⎝
0
0
0
.
.
.
1
⎞
⎟
⎟
⎟
⎟
⎠
(of height
r
)
(3)
in some order. These may be called the
special columns
of
M
. A matrix with some rows
of all zeros is called basic when the rows of all zeros are at the bottom and the submatrix
consisting of the nonzero rows is basic.
All matrices in reduced echelon form are basic matrices. Other examples of e.g. 3
×
7
basic matrices are
⎛
⎝
2
3
0
4
1
0
5
6
7
1
8
0
0
9
5
5
0
5
0
1
5
⎞
⎠
,
⎛
⎝
2
1
7
4
1
0
5
6
0
8
8
4
1
9
0
0
0
0
0
0
0
⎞
⎠
,
⎛
⎝
2
1
0
4
1
0
0
6
0
1
8
0
0
1
5
0
0
7
0
1
0
⎞
⎠
.
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 Winter '08
 Aschbacher
 Linear Algebra, Linear Equations, Equations, Matrices, Systems Of Linear Equations, aij, 8m, 8k

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