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Generating a vector space via a T-cyclic subspace

  1. Sep 5, 2013 #1
    I've been thinking about a problem I made up. The solution may be trivial or very difficult as I have not given too much thought to it, but I can't think of an answer of the top of my head.

    Let ## T:V → V ## be a linear operator on a finite-dimensional vector space ##V##. Does there exist a vector ## v \in V ## for which the T-cyclic subspace of ##V## generated by ##v## is ##V##? This is certainly not true in general, since if ##T## is the zero transformation and ##V## has dimension greater than 1 then no T-cyclic subspace will equal ##V##.

    But what about for an arbitrary linear map?

    BiP
     
  2. jcsd
  3. Sep 5, 2013 #2
    Let's just look at a 2-dim space. Let T = [tex]
    \begin{pmatrix}
    0 & 1 \\
    1 & 0\\

    \end{pmatrix}
    [/tex]
    and v = (1,0). Then Tv = (0,1). Clearly v and Tv are a basis for the entire space.
     
  4. Sep 6, 2013 #3
    How can we generalize this for an arbitrary linear map?

    BiP
     
  5. Sep 6, 2013 #4
    Assuming we are in a finite dimensional vector space, of dimension n, you need to pick a T such that [tex]T^n= T [/tex] but if 1 < m < n [tex] T^m \neq T.[/tex]
    If you do have [tex]T^m = T[/tex] for 1 < m < n then the T-cyclic subspace has only dimension m, because that is all the independent vectors you can generate with powers of T.

    Now there is bound to be some specific vector v such that [tex]T^mv = Tv[/tex] even though [tex]T^m \neq T.[/tex] That vector is in fact any eigenvector of [tex]T^m - T.[/tex] You can't generate the entire space V using that kind of vector v. However, there is bound to be a vector u which is not an eigenvector of [tex]T^m - T[/tex] for any 1 < m < n. You can use u to generate the entire space.

    Does every n dimensional vector space have a linear operator which is cyclic of degree n? Yes. Start with my 2 dimensional example and see if you can find something similar in 3 dimensions. Once you get that far, you'll see how to generate such a T in n dimensions.
     
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