aarjaneiro/planetmath huggingface dataset:
triple cross product

triple cross product

The

cross product

of a vector with a cross product is called the

triple cross product

.

The of the triple cross product or

Lagrange’s

is

a

→

×

(

b

→

×

c

→

)

=

(

a

→

⋅

c

→

)

⁢

b

→

-

(

a

→

⋅

b

→

)

⁢

c

→

(“exterior dot far times near minus exterior dot near times far” — this works also when “exterior” is the last ).

The the vectors

b

→

and

c

→

(when these are not

parallel

).

Note that the use of parentheses in the triple cross products is necessary, since the cross product

operation

is not associative (

http://planetmath.org/GeneralAssociativity

), i.e., generally we have

(

a

→

×

b

→

)

×

c

→

≠

a

→

×

(

b

→

×

c

→

)

(for example:

(

i

→

×

i

→

)

×

j

→

=

0

→

but

i

→

×

(

i

→

×

j

→

)

=

-

j

→

when

(

i

→

,

j

→

,

k

→

)

is a

right-handed

orthonormal basis

of

ℝ

3

).  So the system (

http://planetmath.org/AlgebraicSystem

)

(

ℝ

3

,

+

,

×

)

is not a ring.

A direct

consequence

of the is the

Jacobi identity

a

→

×

(

b

→

×

c

→

)

+

b

→

×

(

c

→

×

a

→

)

+

c

→

×

(

a

→

×

b

→

)

=

0

→

,

which is one of the properties making

(

ℝ

3

,

+

,

×

)

a

Lie algebra

.

It follows from the also that

(

a

→

×

b

→

)

×

(

c

→

×

d

→

)

=

(

a

→

⁢

b

→

⁢

d

→

)

⁢

c

→

-

(

a

→

⁢

b

→

⁢

c

→

)

⁢

d

→

where

(

u

→

⁢

v

→

⁢

w

→

)

means the

triple scalar product

of

u

→

,

v

→

and

w

→

.

Title

triple cross product

Canonical name

TripleCrossProduct

Date of creation

2013-03-22 14:15:53

Last modified on

2013-03-22 14:15:53

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

28

Author

pahio (2872)

Entry type

Definition

Classification

msc 15A72

Synonym

vector triple product

Synonym

triple vector product

Related topic

PhysicalVector

Defines

Lagrange’s

formula

Generated on Fri Feb 9 18:33:01 2018 by

LaTeXML
vector product in general vector spaces

vector product in general vector spaces

The

vector product

can be defined in any finite dimensional

vector space

V

with

dim

⁡

V

=

n

. Let

v

1

,

…

,

v

n

be a basis of

V

, we then define the vector product of the vectors

w

1

,

…

,

w

n

-

1

in the following way:

w

1

×

…

×

w

n

-

1

=

∑

j

=

1

n

v

j

⁢

det

⁡

(

w

1

,

…

,

w

n

-

1

,

v

j

)

.

One can easily see that some of the properties of the vector product are the same as in

ℝ

3

:

•

If one of the

w

i

is equal to

0

, then the vector product is

0

.

•

If

w

i

are

linearly dependent

, then the vector product is

0

.

•

In a

Euclidean vector space

w

1

×

…

×

w

n

-

1

is

perpendicular

to all

w

i

.

Title

vector product in general vector spaces

Canonical name

VectorProductInGeneralVectorSpaces

Date of creation

2013-03-22 14:32:32

Last modified on

2013-03-22 14:32:32

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

5

Author

mathwizard (128)

Entry type

Definition

Classification

msc 15A72

Synonym

vector product

Generated on Fri Feb 9 18:33:13 2018 by

LaTeXML
contraction

contraction

Definition

Let

ω

be a smooth

k

-form on a

smooth manifold

M

,
and let

ξ

be a smooth

vector field

on

M

. The

contraction

of

ω

with

ξ

is the smooth

(

k

-

1

)

-form that maps

x

∈

M

to

ω

x

⁢

(

ξ

x

,

⋅

)

.
In other words,

ω

is
point-wise evaluated with

ξ

in the first slot.
We shall denote this

(

k

-

1

)

-form by

ι

ξ

⁢

ω

.
If

ω

is a

0

-form, we set

ι

ξ

⁢

ω

=

0

for all

ξ

.

Properties

Let

ω

and

ξ

be as above. Then
the following properties hold:

1.

For any real number

k

ι

k

⁢

ξ

⁢

ω

=

k

⁢

ι

ξ

⁢

ω

.

2.

For vector fields

ξ

and

η

ι

ξ

+

η

⁢

ω

=

ι

ξ

⁢

ω

+

ι

η

⁢

ω

,

ι

ξ

⁢

ι

η

⁢

ω

=

-

ι

η

⁢

ι

ξ

⁢

ω

,

ι

ξ

⁢

ι

ξ

⁢

ω

=

0

.

3.

Contraction is an anti-derivation

[

1

]

. If

ω

1

is a

p

-form, and

ω

2

is a

q

-form, then

ι

ξ

⁢

(

ω

1

∧

ω

2

)

=

(

ι

ξ

⁢

ω

1

)

∧

ω

2

+

(

-

1

)

p

⁢

ω

1

∧

(

ι

ξ

⁢

ω

2

)

.

References

1

T. Frankel,

Geometry of physics

,
Cambridge University press,
1997.

Title

contraction

Canonical name

Contraction

Date of creation

2013-03-22 13:37:28

Last modified on

2013-03-22 13:37:28

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Definition

Classification

msc 15A75

Classification

msc 58A10

Generated on Fri Feb 9 18:33:15 2018 by

LaTeXML
visualizations of exterior forms

visualizations of exterior forms

There are (relatively) easy ways to visualize low-dimensional

differential forms

[

1

]

:

A 1-form is locally like a stack of papers; given a vector, it returns a number: how many sheets the arrow pierces.

A 2-form takes a pair of arrows and returns the ”area” of the parallelogram they define.

A 3-form takes a triple of arrows and returns the ”volume” of the parallelliped they span. This explains why in three

dimensions

there’s only a one-dimensional space of 3-forms, and why a global

one-form

tells you about orientation.

References

1

Misner, Thorne, and Wheeler, “Gravitation”, Freeman, 1973.

Editorial note:

Descriptions of these with pictures would be nice (especially for helping to visualize

de Rham cohomology

). Maybe they would be better off in an attached entry, though.

Title

visualizations of exterior forms

Canonical name

VisualizationsOfExteriorForms

Date of creation

2013-03-22 15:28:12

Last modified on

2013-03-22 15:28:12

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

6

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A75

Classification

msc 58A10

Related topic

DifferentialForms

Generated on Fri Feb 9 18:33:28 2018 by

LaTeXML
symmetric algebra

symmetric algebra

Let

M

be a module over a

commutative ring

R

. Form the

tensor algebra

T

⁢

(

M

)

over

R

. Let

I

be the ideal of

T

⁢

(

M

)

generated by elements of the form

u

⊗

v

-

v

⊗

u

where

u

,

v

∈

M

.
Then the

quotient algebra

defined by

S

⁢

(

M

)

:=

T

⁢

(

M

)

/

I

is called the

symmetric algebra

over the ring

R

.

Remark.

Let

R

be a field, and

M

a

finite dimensional

vector space

over

R

. Suppose

{

e

1

,

e

2

,

…

,

e

n

}

is a basis of

M

over

R

. Then

T

⁢

(

M

)

is nothing more than a

free algebra

on the basis elements

e

i

. Alternatively, the basis elements

e

i

can be viewed as non-commuting

indeterminates

in the

non-commutative

polynomial ring

R

⁢

⟨

e

1

,

e

2

,

…

,

e

n

⟩

. This then implies that

S

⁢

(

M

)

is

isomorphic

to the “

commutative

” polynomial ring

R

⁢

[

e

1

,

e

2

,

…

,

e

n

]

, where

e

i

⁢

e

j

=

e

j

⁢

e

i

.

Title

symmetric algebra

Canonical name

SymmetricAlgebra

Date of creation

2013-03-22 15:46:23

Last modified on

2013-03-22 15:46:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Definition

Classification

msc 15A78

Generated on Fri Feb 9 18:33:31 2018 by

LaTeXML
matrix logarithm

matrix logarithm

Unlike the scalar

logarithm

, there are no naturally-defined bases for the

matrix logarithm

; therefore, the matrix logarithm is always taken to be the

natural logarithm

. In general, there may be an infinite number of matrices

B

satisfying

exp

⁡

(

B

)

=

A

; these are known as the logarithms of

A

.

As for the scalar natural logarithm, the matrix logarithm can be defined as a

power series

when

A

is a

square matrix

and

||

I

-

A

||

F

<

1

, where

|

|

⋅

|

|

F

is the

Frobenius matrix norm

. The logarithm this formula produces is known as the

principal logarithm

of

A

.

log

⁡

(

A

)

=

-

∑

k

=

1

∞

(

I

-

A

)

k

k

=

log

⁡

(

I

+

X

)

=

∑

k

=

1

∞

(

-

1

)

k

+

1

k

⁢

X

k

Since this

series expansion

does not converge for all

A

, it is not a global

inverse function

for the

matrix exponential

. In particular,

exp

log

⁡

A

=

A

only holds for

||

I

-

A

||

F

<

1

, and

log

⁡

(

exp

A

)

=

A

only holds for

||

A

||

F

<

2

.

There are other, more general methods of calculating the matrix logarithm. For example, see

\htmladdnormallink

An Explicit Formula for the Matrix Logarithmhttp://arxiv.org/abs/math/0410556.

Title

matrix logarithm

Canonical name

MatrixLogarithm

Date of creation

2013-03-22 15:31:22

Last modified on

2013-03-22 15:31:22

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

11

Author

Andrea Ambrosio (7332)

Entry type

Definition

Classification

msc 15A90

Classification

msc 15A99

Related topic

NaturalLogarithm2

Related topic

MatrixFNorm

Related topic

FrobeniusMatrixNorm

Defines

principal logarithm

Generated on Fri Feb 9 18:33:38 2018 by

LaTeXML
an application of Z-matrix in a mobile radio system

an application of Z-matrix in a mobile radio system

The following is an application of

Z-matrix

in wireless communication called power balancing problem.

Consider

n

pairs of mobile users and receiving antennae. For

i

=

1

,

…

,

n

, mobile user

i

transmits radio signal to antenna

i

. Mobile user

i

transmits at power

P

i

. The radio channel
attenuate the signal and user

i

’s signal is received at antenna

i

with power

G

i

⁢

i

⁢

P

i

, where

G

i

⁢

i

denote the channel
gain. The radio signals also interfere each other. At antenna

i

,
the interference due to user

j

has power

G

i

⁢

j

⁢

P

j

. The
receiver noise power at antenna

i

is denoted by

n

i

. The
signal to interference plus noise at receiver

i

is

Γ

i

=

G

i

⁢

i

⁢

P

i

∑

j

≠

i

G

i

⁢

j

⁢

P

j

+

n

i

To guarantee the quality of received signal, it is required that
the signal to interference plus noise ratio

Γ

i

is equal to
a predefined

constant

γ

i

for all

i

. Given

γ

i

,

i

=

1

,

…

,

n

, we want to find

P

1

,

…

,

P

n

such that the
above equation holds for

i

=

1

,

…

,

n

. Let

A

be the

n

×

n

matrix with zero

diagonal

and

(

i

,

j

)

-entry

(

G

i

⁢

j

⁢

γ

i

)

/

G

i

⁢

i

for

i

≠

j

. We want to solve

(

I

-

A

)

⁢

𝐩

=

𝐧

where

𝐩

=

(

P

1

,

…

,

P

n

)

T

is the power vector and

𝐧

=

(

n

i

⁢

γ

i

/

G

i

⁢

i

)

i

=

1

n

. The matrix

I

-

A

is
a Z-matrix, since all

G

i

⁢

j

and

γ

i

are positive
constants. The required power vector is

(

I

-

A

)

-

1

⁢

𝐧

if

I

-

A

is

invertible

. We also required that the

components

of

𝐩

to be positive as power cannot be negative. The
resulting power vector

(

I

-

A

)

-

1

⁢

𝐧

has positive
components if

(

I

-

A

)

-

1

is a non-negative matrix. In such case,

I

-

A

is an

M-matrix

.

Title

an application of Z-matrix in a mobile radio system

Canonical name

AnApplicationOfZmatrixInAMobileRadioSystem

Date of creation

2013-03-22 16:14:16

Last modified on

2013-03-22 16:14:16

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

6

Author

kshum (5987)

Entry type

Application

Classification

msc 15A99

Generated on Fri Feb 9 18:33:41 2018 by

LaTeXML
dual space

dual space

Dual of a

vector space

; dual bases

Let

V

be a vector space over a field

k

. The

dual

of

V

,
denoted by

V

∗

, is the vector space of

linear forms

on

V

, i.e.

linear mappings

V

→

k

.
The

operations

in

V

∗

are defined

pointwise

:

(

φ

+

ψ

)

⁢

(

v

)

=

φ

⁢

(

v

)

+

ψ

⁢

(

v

)

(

λ

⁢

φ

)

⁢

(

v

)

=

λ

⁢

φ

⁢

(

v

)

for

λ

∈

K

,

v

∈

V

and

φ

,

ψ

∈

V

∗

.

V

is

isomorphic

to

V

∗

if and only if the

dimension

of

V

is finite. If not, then

V

∗

has a larger (

infinite

)
dimension than

V

; in other words, the

cardinal

of any basis
of

V

∗

is strictly greater than the cardinal of any basis of

V

.

Even when

V

is finite-dimensional, there is no canonical or

natural
isomorphism

V

→

V

∗

. But on the other hand, a basis

ℬ

of

V

does define a basis

ℬ

∗

of

V

∗

, and moreover a

bijection

ℬ

→

ℬ

∗

. For suppose

ℬ

=

{

b

1

,

…

,

b

n

}

. For each

i

from

1

to

n

, define a

mapping

β

i

:

V

→

k

by

β

i

⁢

(

∑

k

x

k

⁢

b

k

)

=

x

i

.

It is easy to see that the

β

i

are nonzero elements of

V

∗

and are

independent

. Thus

{

β

1

,

…

,

β

n

}

is a basis of

V

∗

, called the

dual basis

of

ℬ

.

The dual of

V

∗

is called the

second dual

or

bidual

of

V

.
There

is

a very simple

canonical injection

V

→

V

∗

∗

,
and it is an

isomorphism

if the dimension of

V

is finite.
To see it, let

x

be any element of

V

and define a mapping

x

′

:

V

∗

→

k

simply by

x

′

⁢

(

ϕ

)

=

ϕ

⁢

(

x

)

.

x

′

is linear by definition, and it is readily verified that the mapping

x

↦

x

′

from

V

to

V

∗

∗

is linear and

injective

.

Dual of a

topological vector space

If

V

is a topological vector space, the

continuous dual

V

′

of

V

is the

subspace

of

V

∗

consisting of
the

continuous

linear forms.

A

normed

vector space

V

is said to be

reflexive

if the natural
embedding

V

→

V

′′

is an isomorphism. For example,
any finite dimensional space is reflexive, and any

Hilbert space

is
reflexive by the

Riesz representation theorem

.

Remarks

Linear forms are also known as

linear functionals

.

Another way in which a linear mapping

V

→

V

∗

can arise is via
a

bilinear form

V

×

V

→

k

.

The notions of duality extend, in part, from vector spaces to modules,
especially

free modules

over

commutative rings

. A related notion is
the duality in projective spaces.

Title

dual space

Canonical name

DualSpace

Date of creation

2013-03-22 12:16:52

Last modified on

2013-03-22 12:16:52

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

15

Author

Daume (40)

Entry type

Definition

Classification

msc 15A99

Synonym

algebraic dual

Synonym

continuous dual

Synonym

dual basis

Synonym

reflexive

Synonym

natural embedding

Synonym

topological dual

Related topic

DualHomomorphism

Related topic

DoubleDualEmbedding

Related topic

BanachSpace

Related topic

Unimodular

Related topic

LinearFunctional

Related topic

BoundedLinearFunctionalsOnLpmu

Generated on Fri Feb 9 18:33:44 2018 by

LaTeXML
dual space separates points

dual space separates points

The following result is a corollary of the

Hahn-Banach theorem

.

Theorem

-

Let

X

be a

normed vector space

. Given a

linearly independent

set

{

x

1

,

…

,

x

n

}

⊂

X

there exist

continuous

linear functionals

f

1

,

…

,

f

n

∈

X

′

such that

f

j

(

x

k

)

=

δ

j

⁢

k

,

1

≤

j

,

k

≤

n

If

x

∈

s

⁢

p

⁢

a

⁢

n

⁢

{

x

1

,

…

,

x

n

}

, then

x

=

∑

j

=

1

n

f

j

⁢

(

x

)

⁢

x

j

.

The above theorem shows that if

f

⁢

(

x

)

=

f

⁢

(

y

)

for every continuous linear functional

f

then

x

=

y

, therefore the

dual space

X

′

separates the points of

X

.

Title

dual space separates points

Canonical name

DualSpaceSeparatesPoints

Date of creation

2013-03-22 17:30:55

Last modified on

2013-03-22 17:30:55

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Corollary

Classification

msc 15A99

Generated on Fri Feb 9 18:33:47 2018 by

LaTeXML
example of trace of a matrix

example of trace of a matrix

Let

A

=

[

2

4

6

8

10

12

14

16

18

]

,

A

′

=

1

2

⁢

A

=

[

1

2

3

4

5

6

7

8

9

]

and

B

=

[

9

8

7

6

5

4

3

2

1

]

then

trace

⁡

(

A

+

B

)

=

trace

⁡

(

A

)

+

trace

⁡

(

B

)

=

(

2

+

10

+

18

)

+

(

9

+

5

+

1

)

=

45

trace

⁡

(

A

)

=

trace

⁡

(

2

⁢

A

′

)

=

2

⋅

trace

⁡

(

A

′

)

=

2

⋅

trace

⁡

[

1

2

3

4

5

6

7

8

9

]

=

2

⋅

(

1

+

5

+

9

)

=

30

Title

example of trace of a matrix

Canonical name

ExampleOfTraceOfAMatrix

Date of creation

2013-03-22 13:33:20

Last modified on

2013-03-22 13:33:20

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

8

Author

Daume (40)

Entry type

Example

Classification

msc 15A99

Generated on Fri Feb 9 18:33:54 2018 by

LaTeXML
generalized Kronecker delta symbol

generalized Kronecker delta symbol

Let

l

and

n

be

natural numbers

such that

1

≤

l

≤

n

.
Further, let

i

k

and

j

k

be natural numbers in

{

1

,

⋯

,

n

}

for all

k

in

{

1

,

⋯

,

l

}

.
Then the

generalized Kronecker delta symbol

, denoted by

δ

j

1

⁢

⋯

⁢

j

l

i

1

⁢

⋯

⁢

i

l

,
is zero if

i

r

=

i

s

or

j

r

=

j

s

for some

r

≠

s

, or if

{

i

1

,

⋯

,

i

l

}

≠

{

j

1

,

⋯

,

j

l

}

as sets.
If none of the above conditions are met, then

δ

j

1

⁢

⋯

⁢

j

l

i

1

⁢

⋯

⁢

i

l

is defined as the sign of the

permutation

that maps

i

1

⁢

⋯

⁢

i

l

to

j

1

⁢

⋯

⁢

j

l

.

From the definition, it follows that when

l

=

1

,
the generalized Kronecker delta symbol reduces to
the traditional delta symbol

δ

j

i

.
Also, for

l

=

n

, we obtain

δ

j

1

⁢

⋯

⁢

j

n

i

1

⁢

⋯

⁢

i

n

=

ε

i

1

⁢

⋯

⁢

i

n

⁢

ε

j

1

⁢

⋯

⁢

j

n

,

δ

j

1

⁢

⋯

⁢

j

n

1

⁢

⋯

⁢

n

=

ε

j

1

⁢

⋯

⁢

j

n

,

where

ε

j

1

⁢

⋯

⁢

j

n

is the

Levi-Civita permutation symbol

.

For any

l

we can write the generalized delta function
as a

determinant

of traditional delta symbols. Indeed,
if

S

⁢

(

l

)

is the

permutation group

of

l

elements

, then

δ

j

1

⁢

⋯

⁢

j

l

i

1

⁢

⋯

⁢

i

l

=

∑

τ

∈

S

⁢

(

l

)

sign

⁢

τ

⁢

δ

j

1

i

τ

⁢

(

1

)

⁢

⋯

⁢

δ

j

l

i

τ

⁢

(

l

)

=

det

⁡

(

δ

j

1

i

1

⋯

δ

j

1

i

l

⋮

⋱

⋮

δ

j

l

i

1

⋯

δ

j

l

i

l

)

.

The first equality follows since the sum one the first line has only one non-zero term; the term for
which

i

τ

⁢

(

k

)

=

j

k

. The second equality follows from the
definition of the determinant.

Title

generalized Kronecker delta symbol

Canonical name

GeneralizedKroneckerDeltaSymbol

Date of creation

2013-03-22 13:31:38

Last modified on

2013-03-22 13:31:38

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Definition

Classification

msc 15A99

Related topic

LeviCivitaPermutationSymbol3

Generated on Fri Feb 9 18:33:58 2018 by

LaTeXML
Iwasawa decomposition

Iwasawa decomposition

Let

A

be an

invertible matrix

with real coefficients. Then there exist unique matrices

O

,

D

,

U

, such that

1.

A

=

O

⁢

D

⁢

U

.

2.

O

is an

orthogonal matrix

.

3.

D

is a

diagonal matrix

with positive

diagonal

entries.

4.

U

is a

unit upper triangular matrix

.

Title

Iwasawa decomposition

Canonical name

IwasawaDecomposition

Date of creation

2013-03-22 14:32:29

Last modified on

2013-03-22 14:32:29

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

7

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 15A99

Related topic

MatrixFactorization

Generated on Fri Feb 9 18:34:07 2018 by

LaTeXML
Kronecker delta

Kronecker delta

The

Kronecker delta

δ

i

⁢

j

is defined as having value 1 when

i

=

j

and 0 otherwise (

i

and

j

are integers). It may also be written as

δ

i

⁢

j

or

δ

j

i

. It is a special case of the

generalized Kronecker delta symbol

.

The delta symbol was first used in print by Kronecker in 1868

[

1

]

.

Example.

The

n

×

n

identity matrix

I

can be written in terms of the Kronecker delta as simply the matrix of the delta,

I

i

⁢

j

=

δ

i

⁢

j

, or simply

I

=

(

δ

i

⁢

j

)

.

References

1

N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.

Title

Kronecker delta

Canonical name

KroneckerDelta

Date of creation

2013-03-22 12:06:23

Last modified on

2013-03-22 12:06:23

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

7

Author

akrowne (2)

Entry type

Definition

Classification

msc 15A99

Related topic

IdentityMatrix

Related topic

LeviCivitaPermutationSymbol3

Generated on Fri Feb 9 18:34:10 2018 by

LaTeXML
linear functional

linear functional

Let

V

be a

vector space

over a field

K

.
A

linear functional

(or

linear form

) on

V

is a

linear mapping

ϕ

:

V

→

K

,
where

K

is thought of as a one-dimensional vector space over itself.

The

collection

of all linear functionals on

V

can be made into a vector space
by defining

addition

and

scalar multiplication

pointwise

;
this vector space is called the

dual space

of

V

.

The term

linear functional

derives from
the case where

V

is a

space of functions

(see the entry on

functionals

(

http://planetmath.org/Functional

)).
Some authors restrict the term to this case.

Title

linear functional

Canonical

name

LinearFunctional

Date of creation

2013-03-22 12:13:54

Last modified on

2013-03-22 12:13:54

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 15A99

Synonym

linear form

Related topic

DualSpace

Related topic

CalculusOfVariations

Related topic

AdditiveFunction2

Related topic

MultiplicativeLinearFunctional

Generated on Fri Feb 9 18:34:12 2018 by

LaTeXML
modules are a generalization of vector spaces

modules are a generalization of vector spaces

A

http://planetmath.org/node/1022

module is the natural

generalization

of a

vector space

, in fact, when working over a field it is just another word for a vector space.

If

M

and

N

are

R

-modules then a

mapping

f

:

M

→

N

is called an

R

-

morphism

(or

homomorphism

) if:

∀

x

,

y

∈

M

:

f

⁢

(

x

+

y

)

=

f

⁢

(

x

)

+

f

⁢

(

y

)

and

∀

x

∈

M

⁢

∀

λ

∈

R

:

f

⁢

(

λ

⁢

x

)

=

λ

⁢

f

⁢

(

x

)

Note as mentioned in the beginning, if

R

is a field, these properties are the defining properties for a

linear transformation

.

Similarly in vector space terminology the image

Im

⁢

f

:=

{

f

⁢

(

x

)

:

x

∈

M

}

and kernel

Ker

⁢

f

:=

{

x

∈

M

:

f

⁢

(

x

)

=

0

N

}

are called the range and

null-space

respectively.

Title

modules are a generalization of vector spaces

Canonical name

ModulesAreAGeneralizationOfVectorSpaces

Date of creation

2013-03-22 13:38:18

Last modified on

2013-03-22 13:38:18

Owner

jgade (861)

Last modified by

jgade (861)

Numerical id

7

Author

jgade (861)

Entry type

Example

Classification

msc 15A99

Generated on Fri Feb 9 18:34:14 2018 by

LaTeXML