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Math Help - A proof regarding rank

  1. #1
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    A proof regarding rank

     rank(\alpha+\beta) \leq rank(\alpha) + rank(\beta)
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  2. #2
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    Quote Originally Posted by szpengchao View Post
     rank(\alpha+\beta) \leq rank(\alpha) + rank(\beta)
    'rank' is a very broad word and has many different meanings in different topics in mathematics.

    What topics are you talking about and what do alpha and beta represent?
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  3. #3
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    linear algebra

    they r linear maps from V to W.
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  4. #4
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    So the "rank" of \alpha is the dimension of \alpha(V) as a subspace of W. If w is in (\alpha+ \beta)(V) then there exist v in V such that (\alpha+ \beta)v= \alpha(v)+ \beta(v) is in W. Certainly that is true if \alpha(v) and \alpha(v) are in W. Do you see why that means that (\alpha+ \beta)V must be a subspace of the direct sum \alpha(V) and \beta(V)?
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  5. #5
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    ok.

    wait. please check my proof:
    so it is equivalent to prove:
     Im(\alpha+\beta)\subset Im(\alpha)\bigoplus Im(\beta)

     \forall w\in Im(\alpha+\beta), \exists \ \ unique \ \ v\in V, \ \ s.t. \ \ (\alpha+\beta)(v)=w \ \ \Rightarrow w\in Im(\alpha)\bigoplus Im(\beta)
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