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Math Help - prove following are equivalent (need help on two implication)

  1. #1
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    prove following are equivalent (need help on two implication)

    \ T :  V \rightarrow  W is a linear transformation, V and W vector spaces
    1) T is injective
    2) T is surjective
    3) Rank(T) = Dim(V)

    3) ---> 1) nullity(T) + Rank(T) = Dim(V) , so 0 = nullity(T) iff T is injective
    2) ----->3) T surjective implies that Dim(W) = Rank(T) (i don't know how to proceed from here)
    1)-----> 2) T injective iff nullity(T) = 0 , which implies Rank(T) = Dim (V), so the number of linearly indep vectors to form a basis for the image = Rank(T), so Rank(T) \le dim(W) ( don't know how to show the inequality other way around)
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  2. #2
    Junior Member
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    there was also the assumption dim(V) = dim (W) so that makes it easier.
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