I have a matrix with column vectors:
A=[a1 a2 a3 a4 a5]; where a1,a2,a3,a4,a5 are column vectors
a1=[0 0 0 0 1]T
a2=[0 0 0 1 1]T
a3=[0 0 1 1 1]T
a4=[0 1 1 1 1]T
a5=[x 1 1 1 1]T
lie bracket [a1 a2]=d/dx {a2}*a1 - d/dx {a1}*a2 =0
What does it imply?
For span{a1,a2} to be involutive lie-bracket [a1,a2] should belong to span{a1,a2}
Does it say span{a1,a2} belongs is involutive?
I advice you de see all possible Lie-bracket. If [a_1,\,a_2]=0 this means you cannot move in the direction [a_1,\,a_2].
@Chaker Jammazi
Thanks for your answer. But what can I say about the involutive property of the span of matrix containing the lie-brackets as its column vectors. Let say they all are zero.
If the span of vector Lie- brackte's is not the dimension of the space we cannot conclude that the set if Lie- brackte's is involutive. See for example the book of A. Isidori ''nonlinear control systems''.
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