Anku posted an Question
April 09, 2021 • 11:04 am 30 points
  • IIT JAM
  • Mathematics (MA)

Please once tell ...

please once tell .... .. . . .. . . . . . . . .. . . . .. . . . .. . . .

3 Answer(s) Answer Now
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  • Alka gupta best-answer

    this theorem is not applicable for all Linear transformation....see this example

    cropped1515668117308922665.jpg
  • Alka gupta

    this theorem is not applicable for all linear transformation......see an example

    cropped4941109752627263312.jpg
  • Alka gupta best-answer

    yes, if between two vector spaces a LT is defined and dim(V) = dim(V') and we want to check it is an isomorphism or not then in this case it is sufficient to check only one-one no need to check onto

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    Deepak singh 1 Best Answer

    These are two different things . Theorem is on vector space not Linear transformation . Refer attached explanation

    cropped8636666518035922754.jpg
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    the thm is saying isomorph. if same dim., what is that?

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    clearly , take any two finite dimensional vector space , if their dimension is same then they will be isomorphic .. eg take R^2 and P1(R) {polynomial of degree atmost 1} .

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    In Anku's question, we have R3->R3 i.e. same dimension so acc. to the thm. it will be isomorphic, right? and if isomorphic then it implies 1-1, onto, & LT, isn't it?

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    check this L.T , it is neither 1-1 nor onto..

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