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Help with definitions in linear algebra

  1. Apr 28, 2007 #1
    Having a matrix, how can I know if the function the matrix is representing is:

    a) Injective
    b) Bijective

    Thanks in advance
     
  2. jcsd
  3. Apr 28, 2007 #2

    quasar987

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    say the map is L:V-->W

    it is injective iff for all w in W, there is at most one v in with with L(v)=w

    it is bijective if it is injective and surjective.

    These are the definitions. If you're having trouble applying them to a specific problem, you should tell us what the exact problem is and what you've attempted (if anything) and then we can help more.
     
  4. Apr 28, 2007 #3

    HallsofIvy

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    Well, you could look at the definitions of those words! Any function, from one set to another is call "injective" if and only if f(x)= f(y) implies x= y: in other words, different members of the domain are mapped to different members of the range. That does NOT imply "surjective"- that something is mapped to every member of the range. For linear transformations, represented by a matrix, injective means that the domain space is mapped one-to-one to a subspace of the range space: that the rank of the matrix is equal to the dimension of the domain space, not necessairily the dimension of the range space.

    In order to be "bijective" a mapping must be both injective and surjective: "one-to-one" and "onto". For a linear transformation represented by a matrix, that means it must be n by n for some positive integer n and have rank n.
     
  5. Apr 28, 2007 #4
    Thanks by your replies

    If I understood well, any matrix with non-zero determinant will be bijective, right?

    I don't quite get what you mean by domain space and range space. I know what a vector space is. Is it related to it?

    Thanks
     
  6. Apr 29, 2007 #5

    quasar987

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    If f:A-->B is a function, then A is called the domain and the subset f(A) of B is called the range. In the case where f is a linear map from one vector space to another, A is the domain space (or domain vector space) and f(A) is the range space.
     
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