aarjaneiro/planetmath huggingface dataset:
vector space over an infinite field is not a finite union of proper subspaces

vector space over an infinite field is not a finite union of proper subspaces

Theorem 1

.

A

vector space

V

over an infinite field

F

cannot be
a finite union of proper

subspaces

of itself.

Proof.

Let

V

=

V

1

∪

V

2

∪

…

∪

V

n

where
each

V

i

is a proper subspace of

V

and

n

>

1

is minimal.
Because

n

is minimal,

V

n

⊄

V

1

∪

V

2

∪

…

∪

V

n

-

1

.

Let

u

∉

V

n

and let

v

∈

V

n

∖

(

V

1

∪

V

2

∪

…

∪

V

n

-

1

)

.

Define

S

=

{

v

+

t

⁢

u

:

t

∈

𝔽

}

. Since

u

∉

V

n

is not the

zero vector

and the field

𝔽

is infinite,

S

must be infinite.

Since

S

⊂

V

=

V

1

∪

V

2

∪

…

∪

V

n

one of the

V

i

must contain infinitely many vectors in

S

.

However, if

V

n

were to contain a vector, other than

v

, from

S

there would
exist non-zero

t

∈

𝔽

such that

v

+

t

⁢

u

∈

V

n

.
But then

t

⁢

u

=

v

+

t

⁢

u

-

v

∈

V

n

and we would have

u

∈

V

n

contrary to the choice of

u

. Thus

V

n

cannot contain
infinitely many elements in

S

.

If some

V

i

,

1

≤

i

<

n

contained two distinct vectors in

S

,
then there would exist distinct

t

1

,

t

2

∈

𝔽

such that

v

+

t

1

⁢

u

,

v

+

t

2

⁢

u

∈

V

i

. But then

(

t

2

-

t

1

)

⁢

v

=

t

2

⁢

(

v

+

t

1

⁢

u

)

-

t

1

⁢

(

v

+

t

2

⁢

u

)

∈

V

i

and we would have

v

∈

V

i

contrary to
the choice of

v

. Thus for

1

≤

i

<

n

,

V

i

cannot contain
infinitely many elements in

S

either.
∎

Title

vector space over an infinite field is not a finite union of proper subspaces

Canonical name

VectorSpaceOverAnInfiniteFieldIsNotAFiniteUnionOfProperSubspaces

Date of creation

2013-03-22 17:29:43

Last modified on

2013-03-22 17:29:43

Owner

loner (106)

Last modified by

loner (106)

Numerical id

9

Author

loner (106)

Entry type

Theorem

Classification

msc 15A03

Generated on Fri Feb 9 18:20:34 2018 by

LaTeXML
AB is conjugate to BA

AB is conjugate to BA

Proposition 1

.

Given

square matrices

A

and

B

where one is

invertible

then

A

⁢

B

is

conjugate

to

B

⁢

A

.

Proof.

If

A

is invertible then

A

-

1

⁢

A

⁢

B

⁢

A

=

B

⁢

A

. Similarly if

B

is invertible then

B

serves to conjugate

B

⁢

A

to

A

⁢

B

.
∎

The result of course applies to any ring

elements

a

and

b

where one is invertible. It also holds for all group elements.

Remark 2

.

This is a partial

generalization

to the observation that the

Cayley table

of an

abelian group

is

symmetric about

the main diagonal. In abelian groups this follows because

A

⁢

B

=

B

⁢

A

. But in

non-abelian groups

A

⁢

B

is only conjugate to

B

⁢

A

. Thus the

conjugacy class

of a group are symmetric about the main diagonal.

Corollary 3

.

If

A

or

B

is invertible then

A

⁢

B

and

B

⁢

A

have the same

eigenvalues

.

This leads to an alternate proof of

A

⁢

B

and

B

⁢

A

being almost

isospectral

. (

http://planetmath.org/ABAndBAAreAlmostIsospectral

) If

A

and

B

are both

non-invertible

, then we restrict to the non-zero

eigenspaces

E

of

A

so that

A

is invertible on

E

. Thus

(

A

⁢

B

)

|

E

is conjugate to

(

B

⁢

A

)

|

E

and so indeed the two transforms have identical non-zero eigenvalues.

Title

AB is conjugate to BA

Canonical name

ABIsConjugateToBA

Date of creation

2013-03-22 16:00:40

Last modified on

2013-03-22 16:00:40

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

4

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 15A04

Generated on Fri Feb 9 18:20:39 2018 by

LaTeXML
additive function

additive function

Definition 1

.

Let

f

:

V

→

R

be a

function

on a

real vector space

V

(more generally we can consider a

vector space

V

over a field

F

).
We say that

f

is

additive

if

f

⁢

(

x

+

y

)

=

f

⁢

(

x

)

+

f

⁢

(

y

)

for all

x

,

y

∈

V

.

If

f

is additive, we find that

1.

f

⁢

(

0

)

=

0

. In fact

f

⁢

(

0

)

=

f

⁢

(

0

+

0

)

=

f

⁢

(

0

)

+

f

⁢

(

0

)

=

2

⁢

f

⁢

(

0

)

.

2.

f

⁢

(

n

⁢

x

)

=

n

⁢

f

⁢

(

x

)

for

n

∈

ℕ

. In fact

f

⁢

(

n

⁢

x

)

=

f

⁢

(

x

)

+

⋯

+

f

⁢

(

x

)

=

n

⁢

f

⁢

(

x

)

.

3.

f

⁢

(

n

⁢

x

)

=

n

⁢

f

⁢

(

x

)

for

n

∈

ℤ

. In fact

0

=

f

⁢

(

0

)

=

f

⁢

(

x

+

(

-

x

)

)

=

f

⁢

(

x

)

+

f

⁢

(

-

x

)

so that

f

⁢

(

-

x

)

=

-

f

⁢

(

x

)

and hence

f

⁢

(

-

n

⁢

x

)

=

-

f

⁢

(

n

⁢

x

)

=

-

n

⁢

f

⁢

(

x

)

.

4.

f

⁢

(

q

⁢

x

)

=

q

⁢

f

⁢

(

x

)

for

q

∈

ℚ

. In fact

q

⁢

f

⁢

(

p

⁢

x

/

q

)

=

f

⁢

(

q

⁢

(

p

⁢

x

/

q

)

)

=

f

⁢

(

p

⁢

x

)

=

p

⁢

f

⁢

(

x

)

so that

f

⁢

(

p

⁢

x

/

q

)

=

p

⁢

f

⁢

(

x

)

/

q

.

This means that

f

is

ℚ

linear.
Quite surprisingly it is possible to show that there exist

additive functions

which are not linear (for example when

V

is a vector space over the field

ℝ

).

Title

additive function

Canonical

name

AdditiveFunction

Date of creation

2013-03-22 16:17:31

Last modified on

2013-03-22 16:17:31

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

9

Author

paolini (1187)

Entry type

Definition

Classification

msc 15A04

Related topic

LinearFunctional

Generated on Fri Feb 9 18:20:41 2018 by

LaTeXML
adjoint endomorphism

adjoint endomorphism

Definition (the bilinear case).

Let

U

be a

finite-dimensional

vector space

over a field

𝕂

, and

B

:

U

×

U

→

𝕂

a

symmetric

,

non-degenerate

bilinear mapping

, for example
a real

inner product

. For an

endomorphism

T

:

U

→

U

we
define the

adjoint

of

T

relative to

B

to be the endomorphism

T

⋆

:

U

→

U

, characterized by

B

⁢

(

u

,

T

⁢

v

)

=

B

⁢

(

T

⋆

⁢

u

,

v

)

,

u

,

v

∈

U

.

It is convenient to identify

B

with a

linear isomorphism

B

:

U

→

U

*

in the sense that

B

⁢

(

u

,

v

)

=

(

B

⁢

u

)

⁢

(

v

)

,

u

,

v

∈

U

.

We then have

T

⋆

=

B

-

1

⁢

T

*

⁢

B

.

To put it another way,

B

gives an

isomorphism

between

U

and
the dual

U

*

, and the
adjoint

T

⋆

is the endomorphism of

U

that corresponds to the
dual homomorphism (

http://planetmath.org/DualHomomorphism

)

T

*

:

U

*

→

U

*

. Here is a

commutative diagram

to
illustrate this idea:

\xymatrix

⁢

U

⁢

\ar

⁢

[

r

]

T

⋆

⁢

\ar

⁢

[

d

]

B

⁢

&

⁢

U

⁢

\ar

⁢

[

d

]

B

⁢

U

*

⁢

\ar

⁢

[

r

]

T

*

⁢

&

⁢

U

*

Relation to the matrix transpose.

Let

𝐮

1

,

…

,

𝐮

n

be a basis of

U

, and let

M

∈

Mat

n

,

n

(

𝕂

)

be the matrix of

T

relative to this basis, i.e.

∑

j

M

i

j

⁢

𝐮

j

=

T

⁢

(

𝐮

i

)

.

Let

P

∈

Mat

n

,

n

(

𝕂

)

denote the matrix of the inner product
relative to the same basis, i.e.

P

i

⁢

j

=

B

⁢

(

𝐮

i

,

𝐮

j

)

.

Then, the representing matrix of

T

⋆

relative to the same basis
is given by

P

-

1

⁢

M

t

⁢

P

.

Specializing further, suppose that the
basis in question is orthonormal, i.e. that

B

⁢

(

𝐮

i

,

𝐮

j

)

=

δ

i

⁢

j

.

Then, the matrix of

T

⋆

is
simply the

transpose

M

t

.

The Hermitian (sesqui-linear) case.

If

T

:

U

→

U

is an endomorphism of a

unitary space

(a

complex
vector space

equipped with a

Hermitian inner product

(

http://planetmath.org/HermitianForm

)). In this setting we can define we define
the Hermitian adjoint

T

⋆

:

U

→

U

by means of the familiar

adjointness

condition

⟨

u

,

T

⁢

v

⟩

=

⟨

T

⋆

⁢

u

,

v

⟩

,

u

,

v

∈

U

.

However, the analogous

operation

at the matrix level is the

conjugate
transpose

. Thus, if

M

∈

Mat

n

,

n

(

ℂ

)

is the matrix of

T

relative to an

orthonormal basis

, then

M

t

¯

is the
matrix of

T

⋆

relative to the same basis.

Title

adjoint endomorphism

Canonical

name

AdjointEndomorphism

Date of creation

2013-03-22 12:29:36

Last modified on

2013-03-22 12:29:36

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

12

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A04

Classification

msc 15A63

Synonym

adjoint

Related topic

Transpose

Defines

Hermitian adjoint

Generated on Fri Feb 9 18:20:43 2018 by

LaTeXML
admissibility

admissibility

Let

k

be a field,

V

a

vector space

over

k

, and

T

:

V

→

V

a

linear operator

. We say that a

subspace

W

of

V

is

T

-

admissible

if

1.

W

is a

T

-

invariant subspace

;

2.

If

f

∈

k

⁢

[

X

]

(See the

polynomial ring

definition

) and

f

⁢

(

T

)

⁢

x

∈

W

, there is a vector

y

∈

W

such that

f

⁢

(

T

)

⁢

x

=

f

⁢

(

T

)

⁢

y

.

Title

admissibility

Canonical

name

Admissibility

Date of creation

2013-03-22 14:05:07

Last modified on

2013-03-22 14:05:07

Owner

gumau (3545)

Last modified by

gumau (3545)

Numerical id

6

Author

gumau (3545)

Entry type

Definition

Classification

msc 15A04

Synonym

admissible

Generated on Fri Feb 9 18:20:45 2018 by

LaTeXML
Botta - Pierce - Watkins theorem

Botta - Pierce - Watkins theorem

Let

𝔽

be an arbitrary field, and let

n

be a

positive integer

. Consider

ℳ

n

⁢

(

𝔽

)

,

the

vector space

of all

n

×

n

matrices over

𝔽

.

Define

•

𝔰

⁢

𝔩

n

⁢

(

𝔽

)

=

{

A

∈

ℳ

n

⁢

(

𝔽

)

:

tr

⁢

(

A

)

=

0

}

,

•

𝒩

=

{

A

∈

ℳ

n

⁢

(

𝔽

)

:

A

⁢

is nilpotent

}

,

•

𝒢

⁢

ℒ

n

⁢

(

𝔽

)

=

{

A

∈

ℳ

n

⁢

(

𝔽

)

:

det

⁡

(

A

)

≠

0

}

.

Notice that

𝔰

⁢

𝔩

n

⁢

(

𝔽

)

is a

linear subspace

of

ℳ

n

⁢

(

𝔽

)

and

𝒩

⊆

𝔰

⁢

𝔩

n

⁢

(

𝔽

)

.

The Botta – Pierce – Watkins

theorem

on linear preservers of the

nilpotent matrices

[

BPW

]

can be formulated as follows.

Theorem 1

Let

φ

:

s

⁢

l

n

⁢

(

F

)

⟶

s

⁢

l

n

⁢

(

F

)

be a linear

automorphism

. Assume that

φ

⁢

(

N

)

⊆

N

.

Then either

∃

P

∈

G

⁢

L

n

⁢

(

F

)

⁢

∃

c

∈

F

∖

{

0

}

⁢

∀

A

∈

s

⁢

l

n

⁢

(

F

)

:

φ

⁢

(

A

)

=

c

⁢

P

⁢

A

⁢

P

-

1

,

or

∃

P

∈

G

⁢

L

n

⁢

(

F

)

⁢

∃

c

∈

F

∖

{

0

}

⁢

∀

A

∈

s

⁢

l

n

⁢

(

F

)

:

φ

⁢

(

A

)

=

c

⁢

P

⁢

A

T

⁢

P

-

1

.

The original proof is based on the Gerstenhaber - Serezhkin theorem, some

elementary

algebraic geometry

, and the

fundamental theorem of projective geometry

.

References

BPW

P. Botta, S. Pierce, W. Watkins,

Linear transformations

that preserve the nilpotent matrices,

Pacific J. Math.

104

(No. 1): 39–46 (1983).

Title

Botta - Pierce - Watkins theorem

Canonical name

BottaPierceWatkinsTheorem

Date of creation

2013-03-22 19:20:21

Last modified on

2013-03-22 19:20:21

Owner

kammerer (26336)

Last modified by

kammerer (26336)

Numerical id

6

Author

kammerer (26336)

Entry type

Theorem

Classification

msc 15A04

Related topic

FundamentalTheoremOfProjectiveGeometry

Related topic

GerstenhaberSerezhkinTheorem

Generated on Fri Feb 9 18:20:47 2018 by

LaTeXML
conductor of a vector

conductor of a vector

Let

k

be a field,

V

a

vector space

,

T

:

V

→

V

a

linear transformation

, and

W

a

T

-

invariant subspace

of

V

. Let

x

∈

V

. The

T

-

conductor

of

x

in

W

is the set

S

T

⁢

(

x

,

W

)

containing all

polynomials

g

∈

k

⁢

[

X

]

such that

g

⁢

(

T

)

⁢

x

∈

W

. It happens to be that this set is an ideal of the

polynomial ring

. We also use the term

T

-conductor of

x

in

W

to refer to the

generator

of such ideal.

In the special case

W

=

{

0

}

, the

T

-conductor is called

T

-

annihilator

of

x

.
Another way to define the

T

-conductor of

x

in

W

is by saying that it is a

monic polynomial

p

of lowest degree such that

p

⁢

(

T

)

⁢

x

∈

W

. Of course this polynomial happens to be unique. So the

T

-annihilator of

x

is the monic polynomial

m

x

of lowest degree such that

m

x

⁢

(

T

)

⁢

x

=

0

.

Title

conductor of a vector

Canonical name

ConductorOfAVector

Date of creation

2013-03-22 14:05:19

Last modified on

2013-03-22 14:05:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A04

Synonym

T-conductor

Synonym

conductor

Synonym

annihilator

Synonym

annihilator polynomial

Synonym

conductor polynomial

Generated on Fri Feb 9 18:21:05 2018 by

LaTeXML
cyclic decomposition theorem

cyclic decomposition theorem

Let

k

be a field,

V

a finite dimensional

vector space

over

k

and

T

a

linear operator

over

V

. Call a

subspace

W

⊆

V

T

-

admissible

if

W

is

T

-

invariant

and for any

polynomial

f

⁢

(

X

)

∈

k

⁢

[

X

]

with

f

⁢

(

T

)

⁢

(

v

)

∈

W

for

v

∈

V

, there is a

w

∈

W

such that

f

⁢

(

T

)

⁢

(

v

)

=

f

⁢

(

T

)

⁢

(

w

)

.

Let

W

0

be a proper

T

-

admissible subspace

of

V

. There are non

zero vectors

x

1

,

…

,

x

r

in

V

with respective

annihilator polynomials

p

1

,

…

,

p

r

such that

1.

V

=

W

0

⊕

Z

⁢

(

x

1

,

T

)

⊕

⋯

⊕

Z

⁢

(

x

r

,

T

)

(See the

cyclic subspace

definition)

2.

p

k

divides

p

k

-

1

for every

k

=

2

,

…

,

r

Moreover, the

integer

r

and the

minimal polynomials

(

http://planetmath.org/MinimalPolynomialEndomorphism

)

p

1

,

…

,

p

r

are uniquely determined by (1),(2) and the fact that none of

x

k

is zero.

This is “one of the deepest results in

linear algebra

” (Hoffman & Kunze)

Title

cyclic decomposition theorem

Canonical name

CyclicDecompositionTheorem

Date of creation

2013-03-22 14:05:10

Last modified on

2013-03-22 14:05:10

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 15A04

Synonym

T-admissible

Synonym

T

-admissible

Related topic

CyclicSubspace

Defines

admissible subspace

Generated on Fri Feb 9 18:21:07 2018 by

LaTeXML
cyclic subspace

cyclic subspace

Let

V

be a

vector space

over a field

k

, and

x

∈

V

. Let

T

:

V

→

V

be a

linear transformation

. The

T

-

cyclic subspace

generated by

x

is the smallest

T

-

invariant subspace

which contains

x

, and is denoted by

Z

⁢

(

x

,

T

)

.

Since

x

,

T

⁢

(

x

)

,

…

,

T

n

⁢

(

x

)

,

…

∈

Z

⁢

(

x

,

T

)

, we have that

W

:=

span

⁡

{

x

,

T

⁢

(

x

)

,

…

,

T

n

⁢

(

x

)

,

…

}

⊆

Z

⁢

(

x

,

T

)

.

On the other hand, since

W

is

T

-

invariant

,

Z

⁢

(

x

,

T

)

⊆

W

. Hence

Z

⁢

(

x

,

T

)

is the

subspace

generated by

x

,

T

⁢

(

x

)

,

…

,

T

n

⁢

(

x

)

,

…

In other words,

Z

⁢

(

x

,

T

)

=

{

p

⁢

(

T

)

⁢

(

x

)

∣

p

∈

k

⁢

[

X

]

}

.

Remark

. If

Z

⁢

(

x

,

T

)

=

V

we say that

x

is a

cyclic vector

of

T

.

Title

cyclic subspace

Canonical

name

CyclicSubspace

Date of creation

2013-03-22 14:05:03

Last modified on

2013-03-22 14:05:03

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A04

Classification

msc 47A16

Synonym

cyclic vector subspace

Related topic

CyclicDecompositionTheorem

Related topic

CyclicVectorTheorem

Defines

cyclic vector

Generated on Fri Feb 9 18:21:09 2018 by

LaTeXML
cyclic vector theorem

cyclic vector theorem

Let

K

be a field,

V

a

K

-

vector space

of

dimension

n

,
and

f

:

V

⟶

V

a

linear transformation

. Then

f

has a

cyclic vector

if and only if every linear transformation

g

:

V

⟶

V

which commutes with

f

is a

polynomial

of

K

⁢

[

X

]

evaluated in

f

.

Title

cyclic vector theorem

Canonical name

CyclicVectorTheorem

Date of creation

2013-03-22 14:14:16

Last modified on

2013-03-22 14:14:16

Owner

gumau (3545)

Last modified by

gumau (3545)

Numerical id

5

Author

gumau (3545)

Entry type

Theorem

Classification

msc 15A04

Related topic

CyclicSubspace

Generated on Fri Feb 9 18:21:10 2018 by

LaTeXML
dimension theorem for symplectic complement (proof)

dimension theorem for symplectic complement (proof)

We denote by

V

⋆

the

dual space

of

V

, i.e.,

linear mappings

from

V

to

ℝ

. Moreover, we assume known that

dim

⁡

V

=

dim

⁡

V

∗

for any

vector space

V

.

We begin by showing that the
mapping

S

:

V

→

V

*

,

a

↦

ω

⁢

(

a

,

⋅

)

is an

linear isomorphism

. First, linearity is clear, and since

ω

is

non-degenerate

,

ker

⁡

S

=

{

0

}

, so

S

is injective.
To show that

S

is surjective, we apply the

http://planetmath.org/node/2238

rank-nullity theorem

to

S

, which yields

dim

⁡

V

=

dim

⁡

img

S

.
We now have

img

S

⊂

V

*

and

dim

⁡

img

S

=

dim

⁡

V

∗

.
(The first assertion follows directly from the definition of

S

.)
Hence

img

S

=

V

∗

(see this page (

http://planetmath.org/VectorSubspace

)),
and

S

is a surjection. We have shown that

S

is a linear isomorphism.

Let us next define the mapping

T

:

V

→

W

*

,

a

↦

ω

⁢

(

a

,

⋅

)

.
Applying the

http://planetmath.org/node/2238

rank-nullity theorem to

T

yields

dim

⁡

V

=

dim

⁡

ker

⁡

T

+

dim

⁡

img

T

.

(1)

Now

ker

⁡

T

=

W

ω

and

img

T

=

W

*

.
To see the latter assertion, first note that from the definition of

T

, we
have

img

T

⊂

W

*

. Since

S

is a linear
isomorphism, we also have

img

T

⊃

W

*

.
Then, since

dim

⁡

W

=

dim

⁡

W

*

,
the result follows from equation

1

.

□

Title

dimension

theorem for

symplectic complement

(proof)

Canonical name

DimensionTheoremForSymplecticComplementproof

Date of creation

2013-03-22 13:32:52

Last modified on

2013-03-22 13:32:52

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Proof

Classification

msc 15A04

Generated on Fri Feb 9 18:21:20 2018 by

LaTeXML
dual homomorphism of the derivative

dual homomorphism of the derivative

Let

𝒫

n

denote the

vector space

of

real
polynomials

of degree

n

or less, and let

D

n

:

𝒫

n

→

𝒫

n

-

1

denote the ordinary

derivative

.

Linear forms

on

𝒫

n

can be given in terms of evaluations, and so we introduce the
following notation. For every scalar

k

∈

ℝ

, let

Ev

k

(

n

)

∈

(

𝒫

n

)

*

denote the evaluation

functional

Ev

k

(

n

)

:

p

↦

p

⁢

(

k

)

,

p

∈

𝒫

n

.

Note: the degree superscript matters! For example:

Ev

2

(

1

)

=

2

⁢

Ev

1

(

1

)

-

Ev

0

(

1

)

,

whereas

Ev

0

(

2

)

,

Ev

1

(

2

)

,

Ev

2

(

2

)

are

linearly independent

. Let us consider the

dual homomorphism

D

2

*

, i.e. the

adjoint

of

D

2

. We have the following

relations

:

D

2

*

⁢

(

Ev

0

(

1

)

)

=

-

3

2

⁢

Ev

0

(

2

)

+

2

⁢

Ev

1

(

2

)

-

1

2

⁢

Ev

2

(

2

)

,

D

2

*

⁢

(

Ev

1

(

1

)

)

=

-

1

2

⁢

Ev

0

(

2

)

+

1

2

⁢

Ev

2

(

2

)

.

In other words, taking

Ev

0

(

1

)

,

Ev

1

(

1

)

as the basis of

(

𝒫

1

)

*

and

Ev

0

(

2

)

,

Ev

1

(

2

)

,

Ev

2

(

2

)

as the basis of

(

𝒫

2

)

*

, the matrix that represents

D

2

*

is just

(

-

3

2

-

1

2

2

0

-

1

2

1

2

)

Note the contravariant relationship between

D

2

and

D

2

*

. The
former turns second degree

polynomials

into first degree polynomials,
where as the latter turns first degree evaluations into second degree
evaluations. The matrix of

D

2

*

has 2 columns and 3 rows
precisely because

D

2

*

is a

homomorphism

from a 2-dimensional
vector space to a 3-dimensional vector space.

By contrast,

D

2

will be represented by a

2

×

3

matrix. The

dual basis

of

𝒫

1

is

-

x

+

1

,

x

and the dual basis of

𝒫

2

is

1

2

⁢

(

x

-

1

)

⁢

(

x

-

2

)

,

x

⁢

(

2

-

x

)

,

1

2

⁢

x

⁢

(

x

-

1

)

.

Relative to these bases,

D

2

is represented by the

transpose

of the
matrix for

D

2

*

, namely

(

-

3

2

2

-

1

2

-

1

2

0

1

2

)

This corresponds to the following three relations:

D

2

⁢

[

1

2

⁢

(

x

-

1

)

⁢

(

x

-

2

)

]

=

-

3

2

⁢

(

-

x

+

1

)

-

1

2

⁢

x

D

2

⁢

[

x

⁢

(

2

-

x

)

]

=

2

⁢

(

-

x

+

1

)

+

0

⁢

x

D

2

⁢

[

1

2

⁢

x

⁢

(

x

-

1

)

]

=

-

1

2

⁢

(

-

x

+

1

)

+

1

2

⁢

x

Title

dual homomorphism of the derivative

Canonical name

DualHomomorphismOfTheDerivative

Date of creation

2013-03-22 12:35:28

Last modified on

2013-03-22 12:35:28

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

4

Author

rmilson (146)

Entry type

Example

Classification

msc 15A04

Classification

msc 15A72

Generated on Fri Feb 9 18:21:23 2018 by

LaTeXML
Frobenius theorem on linear determinant preservers

Frobenius theorem on linear determinant preservers

Let

𝔽

be an arbitrary field. Consider

ℳ

n

⁢

(

𝔽

)

, the

vector space

of all

n

×

n

matrices over

𝔽

. Let

𝒢

⁢

ℒ

n

⁢

(

𝔽

)

be the set of all

nonsingular matrices

P

∈

ℳ

n

⁢

(

𝔽

)

.

Definition 1

.

A linear

endomorphism

φ

:

M

n

⁢

(

F

)

⟶

M

n

⁢

(

F

)

is said to be in standard form, if either

∃

P

,

Q

∈

G

⁢

L

n

⁢

(

F

)

⁢

∀

A

∈

M

n

⁢

(

F

)

:

φ

⁢

(

A

)

=

P

⁢

A

⁢

Q

or

∃

P

,

Q

∈

G

⁢

L

n

⁢

(

F

)

⁢

∀

A

∈

M

n

⁢

(

F

)

:

φ

⁢

(

A

)

=

P

⁢

A

⊤

⁢

Q

.

The classical on linear preservers of the

determinant

function

[

GF

]

reads as follows.

Theorem 2

.

If

φ

:

M

n

⁢

(

C

)

⟶

M

n

⁢

(

C

)

is a linear

automorphism

such that

det

⁡

(

φ

⁢

(

A

)

)

=

det

⁡

(

A

)

for all

A

∈

M

n

⁢

(

C

)

, then

φ

is in standard form with

det

⁡

(

P

⁢

Q

)

=

1

.

It is well known that the can be strengthened.

Theorem 3

.

Let

F

be an arbitrary field and let

φ

:

M

n

⁢

(

F

)

⟶

M

n

⁢

(

F

)

be a linear endomorphism. Then the following conditions are

equivalent

:

(i)

det

⁡

(

φ

⁢

(

A

)

)

=

det

⁡

(

A

)

for all

A

∈

M

n

⁢

(

F

)

,

(ii)

φ

is in standard form with

det

⁡

(

P

⁢

Q

)

=

1

.

The above strengthened version of the can be derived from the Dieudonné

theorem

on linear preservers of the

singular matrices

.

References

GF

G. Frobenius,

Über die Darstellung der endlichen Gruppen durch lineare Substitutionen

, Sitzungsber., Preuss. Akad. Wiss., Berlin, 1897 (994–1015).

Title

Frobenius theorem on linear determinant preservers

Canonical name

FrobeniusTheoremOnLinearDeterminantPreservers

Date of creation

2013-03-22 19:19:52

Last modified on

2013-03-22 19:19:52

Owner

kammerer (26336)

Last modified by

kammerer (26336)

Numerical id

7

Author

kammerer (26336)

Entry type

Theorem

Classification

msc 15A04

Classification

msc 15A15

Related topic

DieudonneTheoremOnLinearPreserversOfTheSingularMatrices

Generated on Fri Feb 9 18:21:28 2018 by

LaTeXML
image of a linear transformation

image of a linear transformation

Definition

Let

T

:

V

→

W

be a

linear transformation

. Then the

image

of

T

is the set

Im

⁡

(

T

)

=

{

w

∈

W

∣

w

=

T

⁢

(

v

)

⁢

for some

⁢

v

∈

V

}

=

T

⁢

(

V

)

.

0.0.1

Properties

1.

The

dimension

of

Im

⁡

(

T

)

is called the rank of

T

;

2.

T

is a surjection, if and only if

Im

⁡

(

T

)

=

W

;

3.

Im

⁡

(

T

)

is a

vector subspace

of

W

;

4.

If

L

:

W

→

U

is a linear transformation, then

Im

⁡

(

L

⁢

T

)

=

L

⁢

(

Im

⁡

(

T

)

)

;

Title

image of a linear transformation

Canonical

name

ImageOfALinearTransformation

Date of creation

2013-03-22 13:48:32

Last modified on

2013-03-22 13:48:32

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 15A04

Related topic

RankNullityTheorem

Related topic

KernelOfALinearTransformation

Generated on Fri Feb 9 18:21:29 2018 by

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