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Math Help - True or False?

  1. #1
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    True or False?

    If T1 :Rn→Rm and T2 :Rm→Rk are linear transformations, and if T1 is not onto, then neither is T2 ◦ T1.

    I think it's False b/c if T2 (ran(T1)) = Rk then T2◦T1 is onto? Is there a more generic way for me to say that? THANKS!
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  2. #2
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    Re: True or False?

    Quote Originally Posted by Civy71 View Post
    If T1 :Rn→Rm and T2 :Rm→Rk are linear transformations, and if T1 is not onto, then neither is T2 ◦ T1.

    I think it's False b/c if T2 (ran(T1)) = Rk then T2◦T1 is onto? Is there a more generic way for me to say that?
    Yes, the claim is false, but you need a more specific counterexample, not a more general one. Let n = k = 1 and m = 2, let T1 be an inclusion of R^1 into R^2 and T2 be a projection of R^2 onto R^1. Then T2 ◦ T1 is the identity, which is onto.

    For arbitrary (not necessarily linear) functions, if T2 ◦ T1 is onto than so is T2, and if T2 ◦ T1 is an injection, then so is T1. One should be skeptical about stronger claims.
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