Linear Algebra: Linear Transformation Problem

[Ertosthnes]

Homework Statement

Let T$$\in$$L(V,V). Prove that T$$^{2}$$=0 iff T(V)$$\subset$$n(T).

Homework Equations

dim T(V) + dim n(T) = dim V comes to mind.

The Attempt at a Solution

Honestly, I don't know where to start. I have no idea what I'm doing. My book and my professor are both utterly useless and I'm frustrated at how badly I am failing at this.

 

[HallsofIvy]

Homework Statement

Let T$$\in$$L(V,V). Prove that T$$^{2}$$=0 iff T(V)$$\subset$$n(T).

Homework Equations

dim T(V) + dim n(T) = dim V comes to mind.

The Attempt at a Solution

Honestly, I don't know where to start. I have no idea what I'm doing. My book and my professor are both utterly useless and I'm frustrated at how badly I am failing at this.

Does your book have any definitions? For example does it give a definition of "n(T)"? You did not here, but I am going to assume that n(T) is the null space of T: the set of all vectors, v, such that T(v)= 0. One direction should be obvious. If T(v)= 0, then T²(v)= T(T(v))= T(0)= 0. Now the other way. Suppose that T²(v)= T((T(v))= 0. What does that tell you about the T(V)?

 

[Ertosthnes]

Ah yes. n(T) is the null space of T.

Okay, suppose T$$^{2}$$(V) = T(T(V)) = 0:

Let another vector space W = T(V), so T(W) = 0. Then W $$\subset$$ n(T). Substituting we obtain T(V) $$\subset$$ n(T).

Does that work?

 

[HallsofIvy]

Strictly speaking, either T(W)= {0} or T(v)=0 for all v in W. It might be better to say "if v is in T(V) then v= T(u) for some u in V. Then T(v)= T(T(u))= T²(u).

That proves "If T²= 0, then T(V)$\subset$n(T). But you still need to prove "If T(V)$\subset$n(T) then If T²= 0". Of course, that's pretty easy.