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Math Help - [SOLVED] vectors and matrices

  1. #1
    wizard005
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    [SOLVED] vectors and matrices

    need immediate help

    a) Let T: V V be a linear transformation, where V is a finite dimensional vector space. If dim(range(T))=dim(range(T2)) show that the range and null space of T only have the zero vector in common.
    b)
    Let T: l2 (Zn)--- l2 (Zn) be a linear transformation. Show that T is translation invariant if and only if T(z) = ∑K=0N-1 ak Rk(z) for some a0 ,,aN-1
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  2. #2
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    Quote Originally Posted by wizard005 View Post
    need immediate help

    a) Let T: V V be a linear transformation, where V is a finite dimensional vector space. If dim(range(T))=dim(range(T2)) show that the range and null space of T only have the zero vector in common.
    b)
    Let T: l2 (Zn)--- l2 (Zn) be a linear transformation. Show that T is translation invariant if and only if T(z) = ∑K=0N-1 ak Rk(z) for some a0 ,,aN-1
    a) Do this by contradiction. Suppose the result is false, so there is a nonzero vector z in the range of T (so z=Tx for some x in V), with Tz=0. From that, you should be able to show that the null space of T^2 is bigger than the null space of T. Then use the "rank + nullity" theorem to deduce that the range of T^2 is smaller than the range of T.

    b) This is so badly formatted that I can't read the question. It looks as though it should be saying something like

    Let T: l_2 (Z^n)\to l_2 (Z^n) be a linear transformation. Show that T is translation invariant if and only if T(z) = \sum_{k=0}^{n-1} a_k R_k(z) for some a_0 ,\ldots,a_{n-1}.

    Is that right? If so, what is R_k(z)?
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