An Example
An Example
Clear[a]
a={{1,2,1,2},{2,1,2,1},{3,2,3,2},{3,3,3,3},{5,3,5,3}};
MatrixForm[a]
Out[61]=
1 2 1 2 2 1 2 1 3 2 3 2 3 3 3 3 5 3 5 3
Let's recap from above:
Standard basis for Column Space of A
{Transpose[1 0 1/3 1 1/3], Transpose[0 1 4/3 1 7/3]}
Standard basis for Left Nullspace of A
{Transpose[1 0 0 -7/6 1/2], Transpose[0 1 0 1/6 -1/2],
Transpose[0 0 1 -1/6 -1/2]}
Dimensions add to 5.
Standard basis for Row Space of A
{ [1 0 1 0], [0 1 0 1]}
Standard basis for Nullspace of A
{Transpose[1 0 -1 0], Transpose[0 1 0 -1]}
Dimensions add to 4.
What about the orthogonality?
In[62]:=
cs1={1,0,1/3,1,1/3}
cs2={0,1,4/3,1,7/3}
lns1={1,0,0,-7/6,1/2}
lns2={0,1,0,1/6,-1/2}
lns3={0,0,1,-1/6,-1/2}
Out[62]=
1 1
{1, 0, -, 1, -}
3 3
Out[63]=
4 7
{0, 1, -, 1, -}
3 3
Out[64]=
7 1
{1, 0, 0, -(-), -}
6 2
Out[65]=
1 1
{0, 1, 0, -, -(-)}
6 2
Out[66]=
1 1
{0, 0, 1, -(-), -(-)}
6 2
In[67]:=
cs1.lns1
cs1.lns2
cs1.lns3
cs2.lns1
cs2.lns2
cs2.lns3
Out[67]=
0
Out[68]=
0
Out[69]=
0
Out[70]=
0
Out[71]=
0
Out[72]=
0
In[73]:=
rs1={1,0,1,0}
rs2={0,1,0,1}
ns1={1,0,-1,0}
ns2={0,1,0,-1}
Out[73]=
{1, 0, 1, 0}
Out[74]=
{0, 1, 0, 1}
Out[75]=
{1, 0, -1, 0}
Out[76]=
{0, 1, 0, -1}
In[77]:=
rs1.ns1
rs1.ns2
rs2.ns1
rs2.ns2
Out[77]=
0
Out[78]=
0
Out[79]=
0
Out[80]=
0
Mathematica helps reveal the theory. That's what it's all about, folks!