- #1
samkolb
- 37
- 0
Homework Statement
Let V be a finite dimensional, nonzero complex vector space. Let T be be a linear map on V. Show that V contains invariant subspaces of dimension j for j=1, ..., dim V.
Homework Equations
Since V is complex, V contains an invariant subspace of dimension 1.
The Attempt at a Solution
I started with dim V=3. Then V contains an invariant subspace of dimension 1.
Let U1=span{u1} denote this space, and extend this to a basis for V: V=span{u1,v2,v3}.
What I would like to do is show that span{v2,v3} contains an invariant subspace of dimension 1, span{u2}. Then form the invariant subspace U2=span{u1,u2}.
But I don't know how to show that span{v2,v3} contains an invariant subspace of dimension 1. I'm not sure that it's even true.