GrassmannCalculus`
ComposeSpan  |
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Details and Options
Examples
(2)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In[2]:=
★A;
;
 | ★ℬ |
4
A 1-span of the above four dimensional space is just the basis vectors:
In[3]:=
| ComposeSpan |
e
1
e
2
e
3
e
4
Out[3]=
{,,,}
e
1
e
2
e
3
e
4
This can also be entered in the following manner. Then taking a dot product with a set of scalars gives us a parameterization of the space.
In[4]:=
| ★ |
1
e
1
e
2
e
3
e
4
Out[4]=
{,,,}
e
1
e
2
e
3
e
4
Out[4]=
a+b+c+d
e
1
e
2
e
3
e
4
The following is a parameterization of a subspace.
In[5]:=
| ★ |
1
e
1
e
3
Out[5]=
{,}
e
1
e
3
Out[5]=
a+b
e
1
e
3
We can check that the parameterization is in the subspace.
In[6]:=
e
1
e
3
 | ZeroQ |
Out[6]=
e
1
e
3
e
1
e
3
Out[6]=
True
The following is another parameterization of the same subspace.
In[7]:=
| ★ |
1
e
1
e
3
e
1
e
3
| ★ |
| ZeroQ |
e
1
e
3
Out[7]=
{3-2,+}
e
1
e
3
e
1
e
3
Out[7]=
a(3-2)+b(+)
e
1
e
3
e
1
e
3
Out[7]=
(3a+b)+(-2a+b)
e
1
e
3
Out[7]=
True
The following is a 2-span (a span of bivectors) in a subspace, and a resulting parameterization.
In[8]:=
| ★ |
2
e
1
e
3
e
4
Out[8]=
{⋀,⋀,⋀}
e
1
e
3
e
1
e
4
e
3
e
4
Out[8]=
a⋀+b⋀+c⋀
e
1
e
3
e
1
e
4
e
3
e
4
The span and parameterization of the scalar space is:
In[9]:=
| ★ |
0
e
1
e
3
e
4
Out[9]=
{1}
Out[9]=
a
The following command composes a set of k-spans of the space.
In[10]:=
| ComposeSpan |
e
1
e
2
e
3
e
4
Out[10]=
{{,,,},{⋀⋀,⋀⋀,⋀⋀,⋀⋀}}
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
1
e
2
e
4
e
1
e
3
e
4
e
2
e
3
e
4
This may also be entered as:
In[11]:=
| ★ |
 | ★ |
e
1
e
2
e
3
e
4
Out[11]=
{{,,,},{⋀⋀,⋀⋀,⋀⋀,⋀⋀}}
e
1
e
2
e
3
e
4
e
1
e
2
e
3
e
1
e
2
e
4
e
1
e
3
e
4
e
2
e
3
e
4
The following composes all the k-spans of the subspace.
In[12]:=
| ComposeSpan |
e
1
e
3
e
4
Out[12]=
{{1},{,,},{⋀,⋀,⋀},{⋀⋀}}
e
1
e
3
e
4
e
1
e
3
e
1
e
4
e
3
e
4
e
1
e
3
e
4
This may also be entered as:
In[13]:=
| ★ |
| ★ |
e
1
e
3
e
4
Out[13]=
{{1},{,,},{⋀,⋀,⋀},{⋀⋀}}
e
1
e
3
e
4
e
1
e
3
e
1
e
4
e
3
e
4
e
1
e
3
e
4
The following commands generate symbolic spans in the (sub)space of the given dimension. Here a subspace of dimension 3 is used. The generated symbolic vectors are added to the list of VectorSymbols.
In[14]:=
| ★ |
2
Out[14]=
{⋀,⋀,⋀}
s
1
s
2
s
1
s
3
s
2
s
3
Out[14]=
a⋀+b⋀+c⋀
s
1
s
2
s
1
s
3
s
2
s
3
In[15]:=
| ★ |
| ★ |
Out[15]=
{{,,},{⋀⋀}}
s
1
s
2
s
3
s
1
s
2
s
3
Algorithmic Use of ComposeSpan and ComposeCospan
(1)
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