Basic Examples

Test whether two lists of vectors in
n

span the same subspace:
[◼]
SameSpanQ
[{{1,2,3},{1,2,4}},{{2,4,7},{2,4,9}}]
In[]:=
True
Out[]=

Scope

Vectors in C^n

Test vectors in
n

:
list1={{2,,-,13+5},{1,-,1,-1+6},{1-,2+2,-1+,14-18},{1+,1+2,-,5-3},{1+,1+,-1+,5+},{2+2,2-,2-,-15+9}};​​list2={{2,,-,13+5},{1,-,1,-1+6},{1-,2+2,-1+,14-18}};​​
[◼]
SameSpanQ
[list1,list2]
In[]:=
True
Out[]=
Since there are 6 vectors in the first list and only 3 in the second list, the vectors in the first list must be dependent. Check by computing the rank of the matrix:
MatrixRank[list1]
In[]:=
3
Out[]=
Check that the vectors in the first list are in the span of the vectors in the second list:
RowReduce[Join[Transpose[list1],Transpose[list2],2]]//MatrixForm
In[]:=
1
0
0
2+
-
2
3
3+
2
3
1
0
0
0
1
0
-2
1
3
+
5
3
-
4
3
+
4
3
0
1
0
0
0
1
-
2
3
-1-
5
3
0
0
1
0
0
0
0
0
0
0
0
0
Out[]//MatrixForm=
Check that the vectors in the second list are in the span of the vectors in the first list:
RowReduce[Join[Transpose[list2],Transpose[list1],2]]//MatrixForm
In[]:=
1
0
0
1
0
0
2+
-
2
3
3+
2
3
0
1
0
0
1
0
-2
1
3
+
5
3
-
4
3
+
4
3
0
0
1
0
0
1
-
2
3
-1-
5
3
0
0
0
0
0
0
0
0
0
Out[]//MatrixForm=

Spaces of matrices

The standard basis for the space of 2×4 matrices is:
basis=Map[Partition[#,4]&,IdentityMatrix[8]]
In[]:=
{{{1,0,0,0},{0,0,0,0}},{{0,1,0,0},{0,0,0,0}},{{0,0,1,0},{0,0,0,0}},{{0,0,0,1},{0,0,0,0}},{{0,0,0,0},{1,0,0,0}},{{0,0,0,0},{0,1,0,0}},{{0,0,0,0},{0,0,1,0}},{{0,0,0,0},{0,0,0,1}}}
Out[]=
Test whether the following list of 8 vectors spans the same subspace as
basis
:
list={{{1,2,3,4},{2,1,0,1}},{{-2,3,3,0},{2,1,0,2}},{{0,0,0,0},{0,0,1,0}},{{0,0,0,0},{0,0,0,1}},{{0,0,0,0},{1,0,0,0}},{{0,0,0,0},{0,1,0,0}},{{1,0,0,0},{0,0,0,0}},{{-7,6,6,0},{4,2,0,4}}};
In[]:=
[◼]
SameSpanQ
[list,basis]
In[]:=
False
Out[]=
The
th
8
matrix in the list is a linear combination of the
nd
2
and
th
7
vectors:
RowReduce[Transpose[Flatten/@list]]//MatrixForm
In[]:=
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
2
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
-3
0
0
0
0
0
0
0
0
Out[]//MatrixForm=
Check:
{{-7,6,6,0},{4,2,0,4}}2{{-2,3,3,0},{2,1,0,2}}-3{{1,0,0,0},{0,0,0,0}}
In[]:=
True
Out[]=

Function spaces

Two lists of polynomials that determine the same subspace:
[◼]
SameSpanQ
[{1-
2
x
,x-
3
x
,1-
4
x
},{-1+
2
x
,x-
3
x
,
2
x
-
4
x
},x]
In[]:=
True
Out[]=
The Legendre Polynomials of degree at most
n
span the same subspace as the space of polynomials of degree at most
n
:
[◼]
SameSpanQ
[Table[LegendreP[k,x],{k,0,4}],Table[
k
x
,{k,0,4}],x]
In[]:=
True
Out[]=
Two lists of trigonometric functions that span the same subspace:
Two lists of exponential, trigonometric, and hyperbolic functions that span the same subspace: