 DGImageSpace - Maple Help

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 Sequence DGNullSpace(L, A) DGImageSpace(L, A) Parameters

 L - a procedure, defining a linear transformation  from a vector space $𝒜$ of vectors, forms, tensors etc., to another vector space $ℬ$ of vectors, forms, tensors A - a list of vectors, forms, tensors etc., defining a basis for the vector space $\mathrm{𝒜}$ Description

 • Let   be a linear transformation. The null space of is . The image space of is  for some .
 • The command DGNullSpace(L, A) returns a list of elements of $\mathrm{𝒜}$ which define a basis for the null space of $L.$ The command DGImageSpace(L, A) returns a list of elements of $\mathrm{ℬ}$ which define a basis for the image space of $L.$ Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

Example 1.

Let be a 4-dimensional space, let be the vector space of 1-forms on and let be the vector space of 2-forms on $V$. Fix a 1-form on $V$, and define We find the null space and image space of $L$.

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],V\right)$
 ${\mathrm{frame name: V}}$ (4.1)
 V > $A≔\left[\mathrm{dx1},\mathrm{dx2},\mathrm{dx3},\mathrm{dx4}\right]$
 ${A}{≔}\left[{\mathrm{dx1}}{,}{\mathrm{dx2}}{,}{\mathrm{dx3}}{,}{\mathrm{dx4}}\right]$ (4.2)
 V > $\mathrm{\alpha }≔\mathrm{dx1}$
 ${\mathrm{\alpha }}{≔}{\mathrm{dx1}}$ (4.3)
 V > $L≔\mathrm{\beta }↦\mathrm{\alpha }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&wedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\beta }$
 ${L}{≔}{\mathrm{\beta }}{↦}{\mathrm{DifferentialGeometry}}{:-}{\mathrm{&wedge}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\beta }}\right)$ (4.4)
 V > $\mathrm{DGNullSpace}\left(L,A\right)$
 $\left[{\mathrm{dx1}}\right]$ (4.5)
 V > $\mathrm{DGImageSpace}\left(L,A\right)$
 $\left[{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{,}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{,}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx4}}\right]$ (4.6)

Example 2.

Let be a 3-dimensional space, let be the vector space of covariant rank 2 tensors on $V.$  We define $L$ to be the symmetrization operation, that is, for , define . We find the null space and image space for  $L$.

 V > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3}\right],V\right)$
 ${\mathrm{frame name: V}}$ (4.7)
 V > $L≔T↦\mathrm{SymmetrizeIndices}\left(T,\left[1,2\right],"Symmetric"\right)$
 ${L}{≔}{T}{↦}{\mathrm{Tensor}}{:-}{\mathrm{SymmetrizeIndices}}{}\left({T}{,}\left[{1}{,}{2}\right]{,}{"Symmetric"}\right)$ (4.8)
 V > $A≔\mathrm{GenerateTensors}\left(\left[\left[\mathrm{dx1},\mathrm{dx2},\mathrm{dx3}\right],\left[\mathrm{dx1},\mathrm{dx2},\mathrm{dx3}\right]\right]\right)$
 ${A}{≔}\left[{\mathrm{dx1}}{}{\mathrm{dx1}}{,}{\mathrm{dx1}}{}{\mathrm{dx2}}{,}{\mathrm{dx1}}{}{\mathrm{dx3}}{,}{\mathrm{dx2}}{}{\mathrm{dx1}}{,}{\mathrm{dx2}}{}{\mathrm{dx2}}{,}{\mathrm{dx2}}{}{\mathrm{dx3}}{,}{\mathrm{dx3}}{}{\mathrm{dx1}}{,}{\mathrm{dx3}}{}{\mathrm{dx2}}{,}{\mathrm{dx3}}{}{\mathrm{dx3}}\right]$ (4.9)

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

 V > $\mathrm{DGNullSpace}\left(L,A\right)$
 $\left[{\mathrm{dx1}}{}{\mathrm{dx2}}{-}{\mathrm{dx2}}{}{\mathrm{dx1}}{,}{\mathrm{dx1}}{}{\mathrm{dx3}}{-}{\mathrm{dx3}}{}{\mathrm{dx1}}{,}{\mathrm{dx2}}{}{\mathrm{dx3}}{-}{\mathrm{dx3}}{}{\mathrm{dx2}}\right]$ (4.10)

and the image space is the space of symmetric tensors.

 V > $\mathrm{DGImageSpace}\left(L,A\right)$
 $\left[{\mathrm{dx1}}{}{\mathrm{dx1}}{,}\frac{{\mathrm{dx1}}}{{2}}{}{\mathrm{dx2}}{+}\frac{{\mathrm{dx2}}}{{2}}{}{\mathrm{dx1}}{,}\frac{{\mathrm{dx1}}}{{2}}{}{\mathrm{dx3}}{+}\frac{{\mathrm{dx3}}}{{2}}{}{\mathrm{dx1}}{,}{\mathrm{dx2}}{}{\mathrm{dx2}}{,}\frac{{\mathrm{dx2}}}{{2}}{}{\mathrm{dx3}}{+}\frac{{\mathrm{dx3}}}{{2}}{}{\mathrm{dx2}}{,}{\mathrm{dx3}}{}{\mathrm{dx3}}\right]$ (4.11)