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Math Help - Linear Map

  1. #1
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    Linear Map

    Hello,

    (a) Does there exist a linear map T : R^11-------->R^5 such that rank(T)=null(T)?

    My attempt:Never because rank(T)+null(T)=dim(R^11)=11 , we obtain 2rank(T)=11, contradicting the fact that rank(T) is an integer.

    (b) Does there exist a linear map T : R^6-------->R^2 such that 2rank(T) = nul(T)?

    Yes. rank(T)+null(T)=dimm(R^6)=6
    3rank(T)=6 so rank(T)=2

    are these correct?

    Thanks
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  2. #2
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    Quote Originally Posted by charikaar View Post
    Hello,

    (a) Does there exist a linear map T : R^11-------->R^5 such that rank(T)=null(T)?

    My attempt:Never because rank(T)+null(T)=dim(R^11)=11 , we obtain 2rank(T)=11, contradicting the fact that rank(T) is an integer.

    (b) Does there exist a linear map T : R^6-------->R^2 such that 2rank(T) = nul(T)?

    Yes. rank(T)+null(T)=dimm(R^6)=6
    3rank(T)=6 so rank(T)=2

    are these correct?

    Thanks
    Basically yes, but I'd require from a student to produce a specific example in (b), and not only to show that it is possible.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Basically yes, but I'd require from a student to produce a specific example in (b), and not only to show that it is possible

    Tonio
    can you help me with example please? thanks
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  4. #4
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    Quote Originally Posted by charikaar View Post
    can you help me with example please? thanks

    Any onto map \mathbb{R}^6 \rightarrow \mathbb{R}^2 will do. I bet you can come up with at least 2 very simple such maps.

    Tonio
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  5. #5
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    is this correct





    Thanks
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  6. #6
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    Quote Originally Posted by charikaar View Post
    is this correct





    Thanks

    No, of course i'ts not right. You have to produce a map from \mathbb{R}^6\,\,to\,\,\mathbb{R}^2, and your map above is defined only on two lin. indep. vectors in \mathbb{R}^6 and besides this it maps to... \mathbb{R}^6 again!

    Tonio
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