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Rank 193
Rank
The number of linearly independent vectors among the columns of the matrix M
(which is also the dimension of the R
m
subspace Im f ) is called the rank of M, and
it is written like this: rank M.
Example 1
The linear system of equations , that is ,
can be rewritten as follows:
The two vectors and are linearly independent, as can be seen on
pages 133 and 135, so the rank of is 2.
Also note that
y
1
y
2
=
3x
1
+ 1x
2
1x
1
+ 2x
2
3
1
1
2
det = 3 · 2 − 1 · 1 = 5 ≠ 0.
3
1
1
2
3x
1
+ 1x
2
= y
1
1x
1
+ 2x
2
= y
2
3
1
1
2
y
1
y
2
= =
= x
1
x
1
x
2
3
1
1
2
3x
1
+ 1x
2
1x
1
+ 2x
2
3
1
+ x
2
1
2
Example 2
The linear system of equations , that is ,
can be rewritten as follows:
So the rank of is 1.
Also note that
y
1
y
2
=
3x
1
+ 6x
2
1x
1
+ 2x
2
3x
1
+ 6x
2
= y
1
1x
1
+ 2x
2
= y
2
y
1
y
2
= =
x
1
x
2
3
1
6
2
3x
1
+ 6x
2
1x
1
+ 2x
2
3
1
6
2
det = 3 · 2 − 6 · 1 = 0.
3
1
6
2
= x
1
3
1
+ x
2
6
2
= [x
1
+ 2x
2
]
3
1
= x
1
3
1
+ 2x
2
3
1
194 Chapter 7 Linear Transformations
Example 3
The linear system of equations , that is ,
can be rewritten as:
The two vectors and are linearly independent, as we discovered
on page 137, so the rank of is 2.
The system could also be rewritten like this:
Note that
=
y
1
y
2
y
3
1x
1
+ 0x
2
0x
1
+ 1x
2
0x
1
+ 0x
2
1x
1
+ 0x
2
= y
1
0x
1
+ 1x
2
= y
2
0x
1
+ 0x
2
= y
3
1
0
0
0
1
0
1
0
0
0
1
0
= 0. det
1
0
0
0
1
0
0
0
0
= x
1
x
1
x
2
+ x
2
=
y
1
y
2
y
3
=
1
0
0
0
1
0
1
0
0
0
1
0
1x
1
+ 0x
2
0x
1
+ 1x
2
0x
1
+ 0x
2
=
y
1
y
2
y
3
x
1
x
2
x
3
=
1
0
0
0
1
0
0
0
0
1x
1
+ 0x
2
0x
1
+ 1x
2
0x
1
+ 0x
2
Rank 195
Example 4
The linear system of equations , that is
, can be rewritten as follows:
The rank of is equal to 2, as we’ll see on page 203.
The system could also be rewritten like this:
Note that .
The four examples seem to point to the fact that
This is indeed so, but no formal proof will be given in this book.
a
12
a
22
a
n2
a
1n
a
2n
a
nn
a
11
a
21
a
n1
a
12
a
22
a
n2
a
1n
a
2n
a
nn
a
11
a
21
a
n1
det = 0 is the same as rank ≠ n.
=
y
1
y
2
1x
1
+ 0x
2
+ 3x
3
+ 1x
4
0x
1
+ 1x
2
+ 1x
3
+ 2x
4
1x
1
+ 0x
2
+ 3x
3
+ 1x
4
= y
1
0x
1
+ 1x
2
+ 1x
3
+ 2x
4
= y
2
1
0
0
1
3
1
1
2
1
0
0
0
0
1
0
0
3
1
0
0
1
2
0
0
x
1
x
2
x
3
x
4
=
y
1
y
2
y
3
y
4
1x
1
+ 0x
2
+ 3x
3
+ 1x
4
0x
1
+ 1x
2
+ 1x
3
+ 2x
4
0
0
=
=
y
1
y
2
1x
1
+ 0x
2
+ 3x
3
+ 1x
4
0x
1
+ 1x
2
+ 1x
3
+ 2x
4
=
1
0
0
1
3
1
1
2
x
1
x
2
x
3
x
4
= x
1
1
0
+ x
2
0
1
+ x
3
3
1
+ x
4
1
2
= 0det
1
0
0
0
0
1
0
0
3
1
0
0
1
2
0
0

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