Linear Algebra: Linear Transformation Problem

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  • #1
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Homework Statement


Let T[tex]\in[/tex]L(V,V). Prove that T[tex]^{2}[/tex]=0 iff T(V)[tex]\subset[/tex]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.
 

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  • #2
HallsofIvy
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Homework Statement


Let T[tex]\in[/tex]L(V,V). Prove that T[tex]^{2}[/tex]=0 iff T(V)[tex]\subset[/tex]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)?
 
  • #3
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Ah yes. n(T) is the null space of T.

Okay, suppose T[tex]^{2}[/tex](V) = T(T(V)) = 0:

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

Does that work?
 
  • #4
HallsofIvy
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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)[itex]\subset[/itex]n(T). But you still need to prove "If T(V)[itex]\subset[/itex]n(T) then If T2= 0". Of course, that's pretty easy.
 

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