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Math Help - inner product spaces

  1. #1
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    Question inner product spaces

    Can you please, help me with this
    Let T: V->W is linear. Prove that
    (a) T is injective if and only it T* is surjective.
    (b) T is surjective if and only it T* is injective.
    Thank you!
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  2. #2
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    Quote Originally Posted by bamby View Post
    Can you please, help me with this
    Let T: V->W is linear. Prove that
    (a) T is injective if and only it T* is surjective.
    (b) T is surjective if and only it T* is injective.
    Thank you!
    Start with the definitions. What is the definition of "injective"? What is the definition of "surjective"? What is the definition of A*?
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  3. #3
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    inner product spaces

    I am so stupid but I don't see how if T is injective will get T* is surjective. And if T is surjective T* will be injective.
    And by A* do you mean the transition matrix?
    Thank you!
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  4. #4
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    Start with the definition of T*: \langle Tx,y\rangle = \langle x,T^*y\rangle (where the angled brackets denote the inner product).

    Now ask yourself what does it mean to say that \langle Tx,y\rangle = \langle x,T^*y\rangle = 0 for all vectors y. If the first of those inner products is zero for all y it tells you that Tx must be zero. If the second inner product is zero for all y it tells you that x must be orthogonal (perpendicular) to the range of T*. You should be able to see from this that the kernel of T is {0} if and only if the range of T* is the whole space.
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