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DifferentialGeometry[DGNullSpace] - find the null space of a linear transformation acting on a vector space of vectors, differential forms, tensors

DifferentialGeometry[DGImageSpace] - find the image space of a linear transformation acting on a vector space of vectors, differential forms, tensors

 

Calling Sequences

     DGNullSpace(L, A)

     DGImageSpace(L, A)

Parameters

     L    - a procedure, defining a linear transformation L:𝒜 ℬ  from a vector space 𝒜 of vectors, forms, tensors etc., to another vector space  of vectors, forms, tensors

     A        - a list of vectors, forms, tensors etc., defining a basis for the vector space 𝒜

     

 

Description

Examples

Description

• 

Let  L:𝒜   be a linear transformation. The null space of L is NL = {a  𝒜 | La = 0}. The image space of L is ImL = {b  ℬ |  b = La for some a  𝒜}.

• 

The command DGNullSpace(L, A) returns a list of elements of 𝒜 which define a basis for the null space of L. The command DGImageSpace(L, A) returns a list of elements of ℬ which define a basis for the image space of L.

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

Let V be a 4-dimensional space, let 𝒜 be the vector space of 1-forms on V, and let ℬ  be the vector space of 2-forms on V. Fix a 1-form α on V, and define Lβ = α β.  We find the null space and image space of L.

 

DGsetupx1,x2,x3,x4,V

frame name: V

(2.1)
V > 

Adx1,dx2,dx3,dx4

Adx1,dx2,dx3,dx4

(2.2)
V > 

αdx1

αdx1

(2.3)
V > 

Lβ→α &wedge β

LβDifferentialGeometry:−&wedgeα,β

(2.4)
V > 

DGNullSpaceL,A

dx1

(2.5)
V > 

DGImageSpaceL,A

dx1dx2,dx1dx3,dx1dx4

(2.6)

 

Example 2.

Let V be a 3-dimensional space, let 𝒜  be the vector space of covariant rank 2 tensors on V.  We define L to be the symmetrization operation, that is, for T  𝒜, define LTX, Y =  1/2 TX,Y + TY,X. We find the null space and image space for  L.

V > 

DGsetupx1,x2,x3,V

frame name: V

(2.7)
V > 

LT→SymmetrizeIndicesT,1,2,Symmetric

LTTensor:−SymmetrizeIndicesT,1,2,Symmetric

(2.8)
V > 

AGenerateTensorsdx1,dx2,dx3,dx1,dx2,dx3

Adx1dx1,dx1dx2,dx1dx3,dx2dx1,dx2dx2,dx2dx3,dx3dx1,dx3dx2,dx3dx3

(2.9)

 

The null space of L is the space of skew-symmetric tensors,

V > 

DGNullSpaceL,A

dx1dx2dx2dx1,dx1dx3dx3dx1,dx2dx3dx3dx2

(2.10)

 

and the image space is the space of symmetric tensors.

V > 

DGImageSpaceL,A

dx1dx1,dx12dx2+dx22dx1,dx12dx3+dx32dx1,dx2dx2,dx22dx3+dx32dx2,dx3dx3

(2.11)

See Also

DifferentialGeometry

Annihilator

ComplementaryBasis

DGbasis

DGsolve

IntersectSubspaces

 


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