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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