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Don't understand this simple vector space problem

  1. May 17, 2007 #1
    Don't understand this reasoning with respect to linear operators.

    Let S and T be linear operators on the finite dimensional vector space V. Then assuming the composition ST is invertible, we get
    [tex]\text{null} \; S \subset \text{null} \; ST [/tex]

    Why is that? I thought hard about it but I simply cannot follow. Is it not possible to have an element x of V that is in the nullspace of S but not in the nullspace of ST ? i.e. S maps x to 0 but T maps x to y where S don't map y to 0 ?
     
  2. jcsd
  3. May 17, 2007 #2

    matt grime

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    If ST is invertible, then the null space of it is zero. I think you ought to re-examine your question.

    It is certainly true that null(T) is a subspace of null(ST).
     
  4. May 17, 2007 #3
    yes I understand that the null space of ST i zero, and that null(T) is a subspace of null(ST), but how can we say that null(S) is a subspace of null(ST) ? I must be doing some kind of faulty thinking :(
     
  5. May 17, 2007 #4

    matt grime

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    You can't say null(S) is a subspace of null(ST), in general. It isn't. Trivially. Howver, you asserted that ST was invertible, and at no point attempted to use this fact. Thus null(ST)=0, so you're asking 'is null(S) a subspace of the zero vector space'. Well, what is the only subspace of the zero space? I.e. is S injective?
     
  6. May 18, 2007 #5

    mathwonk

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    it depends which convention on composition is being used. some people write ST for first S then T, but not me.
     
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