Show that the plane x^2 + x^3 = 0 is an invariant subspace for the matrix
A = 1 2 3
1 2 1
1 1 2
What do you mean by x^2, x^3 ?
If You acquire the notation:
(x^1, x^2, x^3)
for a generic element of Your space than the condition
x^2 + x^3 = 0
means that the generic element of given subspace is of the form
x^1(1,0,0)+x^2(0,1,-1)
Observe that A*(1,0,0) = (1,1,1) which is not an element of our subspace...
So, explain the notation or give us some clue
Well, I don't see how does this equation gives a plane... It gives us two parallel planes x=0, x=-1 neither of which is invariant subspace of A. Probably I still don't understand something.
I've computed invariant subspaces of A - maybe this will help.
(1) spanned by a vector (1,-3,2)
(2) '' (-1-Sqrt[6], 1,1)
(3) '' (-1+Sqrt[6], 1,1)
Good luck
magda