change of basis

change of basis

Let

V

be a

vector space

. Given a basis

A

for

V

, each vector

v

∈

V

can be uniquely expressed in terms of the base

elements

v

i

∈

A

as follows:

v

=

∑

v

i

∈

A

r

i

⁢

v

i

where the sum is taken over a finite number of elements in

A

. Suppose now that

B

is another basis for

V

. By a

change of basis

from

A

to

B

we mean re-expressing

v

in terms of base elements

w

i

∈

B

.

Formally, we can think of a change of basis as the

identity function

(viewed as a

linear operator

) on a vector space

V

, such that elements in the domain are expressed in terms of

A

and elements in the range are expressed in terms of

B

.

Note that, by the very design of a basis, a change of basis in a vector space is always possible.

Now, if

V

has

dimension

n

<

∞

. We can total order bases

A

and

B

. Then a change of basis (from

A

to

B

) has the

matrix representation

[

I

]

B

A

,

where

I

:

V

→

V

is the identity operator.

[

I

]

B

A

is called a

change of basis matrix

. By applying

[

I

]

B

A

to a vector

v

expressed in terms of

A

, we get

v

expressed in terms of

B

:

[

v

]

B

=

[

I

]

B

A

⁢

[

v

]

A

,

where

[

v

]

A

and

[

v

]

B

are

v

expressed in the two bases

A

and

B

respectively.

Since

I

is obviously

invertible

,

[

I

]

B

A

is invertible also, whose

inverse

is

[

I

]

A

B

. Furthermore,

[

I

]

A

=

I

n

for any basis

A

. Here,

I

n

is the

identity matrix

.

Examples

.

1.

Let

V

=

ℝ

3

and the following two sets

A

=

{

(

1

-

1

2

)

,

(

0

1

1

)

,

(

2

3

0

)

}

⁢

and

⁢

B

=

{

(

1

0

0

)

,

(

0

1

0

)

,

(

0

0

1

)

}

be the two ordered bases for

V

, ordered in the way the elements are arranged in the set. For each

v

i

∈

A

,

I

⁢

(

v

i

)

=

v

i

=

[

v

i

]

E

3

, we see that

[

I

]

B

A

=

(

1

0

2

-

1

1

3

2

1

0

)

.

Notice that the columns of

[

I

]

B

A

are exactly the elements of

A

. Indeed, each element of

A

is

already

written in terms of the

standard basis

elements (in

B

). For example, let

v

be the first basis element in

A

. Let us see what

[

v

]

A

is, when expressed using base elements in

B

, the standard ordered basis:

[

v

]

B

=

[

I

]

B

A

⁢

[

v

]

A

=

(

1

0

2

-

1

1

3

2

1

0

)

⁢

[

v

]

A

=

(

1

0

2

-

1

1

3

2

1

0

)

⁢

(

1

0

0

)

=

(

1

-

1

2

)

,

exactly as we have expected.

2.

Conversely

, let

w

be the first basis element in

B

. What is

w

when expressed in terms of basis elements of

A

? In other words, we need to find

[

w

]

A

=

[

I

]

A

B

⁢

[

w

]

B

.

Now,

[

w

]

B

is just

(

1

0

0

)

, so

[

w

]

A

is nothing more than the first column of

[

I

]

A

B

, which is just the inverse of the matrix

[

I

]

B

A

, so

[

I

]

A

B

=

(

[

I

]

B

A

)

-

1

=

(

1

0

2

-

1

1

3

2

1

0

)

-

1

=

(

1

/

3

-

2

/

9

2

/

9

-

2

/

3

4

/

9

5

/

9

1

/

3

1

/

9

-

1

/

9

)

.

Therefore,

[

w

]

A

=

(

1

/

3

-

2

/

3

1

/

3

)

. A quick verification shows that this is indeed the case:

(

1

0

0

)

=

(

1

/

3

)

⁢

(

1

-

1

2

)

+

(

-

2

/

3

)

⁢

(

0

1

1

)

+

(

1

/

3

)

⁢

(

2

3

0

)

.

3.

Now let

C

be the set

{

(

1

0

2

)

,

(

0

1

1

)

,

(

2

1

0

)

}

. It is easy to check that

C

forms a basis for

ℝ

3

(

determinant

is non-zero). Order

C

in the

obvious

manner. What is the change of basis matrix

[

I

]

A

C

? One way is to express each element of

C

in terms of the elements of

A

. Another way is to use the

formula

[

I

]

A

C

=

[

I

]

A

B

⁢

[

I

]

B

C

. Applying the first example, we see that

[

I

]

B

C

is just the matrix whose columns are elements of

C

. As a result:

[

I

]

A

C

=

[

I

]

A

B

⁢

[

I

]

B

C

=

(

1

/

3

-

2

/

9

2

/

9

-

2

/

3

4

/

9

5

/

9

1

/

3

1

/

9

-

1

/

9

)

⁢

(

1

0

2

0

1

1

2

1

0

)

=

(

7

/

9

0

4

/

9

4

/

9

1

-

8

/

9

1

/

9

0

7

/

9

)

.

Remarks.

Let us summarize what we have learned from the examples above, as well as list some additional facts. Let

V

be a finite dimensional vector space of dimension

n

.

•

If

E

is the standard basis (ordered), then for any ordered basis

A

,

[

I

]

E

A

is the matrix whose columns are exactly the basis elements in

A

(assuming these elements have already been expressed in terms of

E

) such that the

i

-column corresponds to the

i

-th element in the ordered set

A

.

•

This also means that every

invertible matrix

A

corresponds to (in a

one-to-one

fashion) a change of basis from the basis

S

A

whose elements are columns of

A

to

E

, the standard basis:

A

=

[

I

]

E

S

A

.

•

Continue to assume that

E

is the standard basis. Let

A

,

B

be any ordered bases for

V

. Using the above

property

, we can easily compute

[

I

]

B

A

, which is

[

I

]

B

E

⁢

[

I

]

E

A

=

(

[

I

]

E

B

)

-

1

⁢

[

I

]

E

A

.

•

Let

A

′

be a re-ordering of the ordered basis

A

, where each

v

i

′

∈

A

′

is just

v

π

⁢

(

i

)

for some

permutation

in

S

n

. Then

[

I

]

A

A

′

is the

permutation matrix

corresponding to the permutation

π

.

•

Suppose

T

is a linear transformation from

V

to

W

(both finite dimensional). Under a bases

A

⊂

V

and

B

⊂

W

,

T

has matrix representation

[

T

]

B

A

. Under changes of basis from

A

to

A

′

, and

B

to

B

′

, we have

[

T

]

B

′

A

′

=

[

I

⁢

T

⁢

I

]

B

′

A

′

=

[

I

]

B

′

B

⁢

[

T

]

B

A

⁢

[

I

]

A

A

′

.

•

If

T

is a linear operator on

V

, then setting

V

=

W

,

A

=

B

and

A

′

=

B

′

from above, we have that

[

T

]

A

′

=

P

-

1

⁢

[

T

]

A

⁢

P

,

where

P

is the change of basis matrix

[

I

]

A

A

′

. This shows that

[

T

]

A

and

[

T

]

A

′

are

similar matrices

. In other words, under a change of basis, the linear transformation

T

is basically the same.

References

1

Friedberg, Insell, Spence.

Linear Algebra

. Prentice-Hall Inc., 1997.

Title

change of basis

Canonical

name

ChangeOfBasis

Date of creation

2013-03-22 17:30:18

Last modified on

2013-03-22 17:30:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

20

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A04

Synonym

change of coordinates

Synonym

change of bases

Synonym

basis change

Synonym

base change

Synonym

base change matrix

Defines

change of basis matrix

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