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Math Help - Isomorphism

  1. #1
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    Isomorphism

    Show that two finite dimensional vector spaces over F are isomorphic iff. they have equal dimensions.

    (\Rightarrow)
    Suppose X, Y are isomorphic where X, Y are finite dimensional vector spaces.
    Since f is isomorphic, f is monic and an epimorphism. Therefore, f is in a one to one correspondence. Thus, dim X = dim Y.

    (\Leftarrow)
    Suppose dim X = dim Y.
    Define f:X\to Y by f(x_i)=y_i.

    If f(x_1)=f(x_2), then
    f(x_1)=y_1 \ \text{and} \ f(x_2)=y_2\Rightarrow y_1=y_2
    Therefore, f is monic.

    There exist y_i\in Y such that y_i=f(x_i). Therefore, f is an epimorphism.

    Hence, f is an isomorphism.

    Correct?
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  2. #2
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    Re: Isomorphism

    you haven't defined what the x_i and the y_i ARE.

    what is the dimension of a vector space? it is the cardinality of any basis. therefore, bases should figure prominently in your proof. i don't see the word "basis" once.

    for example, to prove that two isomorphic vector spaces X,Y have the same dimension, one way is to pick a basis {x1,x2,...xn} for X.

    since we have an isomorphism f, show that {f(x1),f(x2),...,f(xn)} is a basis for Y. we know that it has the same cardinality, because f is bijective,

    so linear independence/spanning is the only thing to prove.

    conversely, if X,Y have the same dimension, you need to show that a map which maps a basis {x1,x2,...,xn} for X bijectively to a basis {y1,y2,...,yn} for Y,

    can be extended to an isomorphism of X with Y.

    this map must be surjective (a consequence of the {yi} being a spanning set), injective (a consequence of the {xi} being linearly independent),

    and linear (which suggests that we extend our basis map linearly). even if you merely failed to specify that the {xi} and {yi} are bases,

    showing that f is bijective just on those sets,

    does not make f into a linear map on all of X. you need to say HOW you are defining f on an arbitrary element of X.
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  3. #3
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    Re: Isomorphism

    Quote Originally Posted by Deveno View Post
    you haven't defined what the x_i and the y_i ARE.

    what is the dimension of a vector space? it is the cardinality of any basis. therefore, bases should figure prominently in your proof. i don't see the word "basis" once.

    for example, to prove that two isomorphic vector spaces X,Y have the same dimension, one way is to pick a basis {x1,x2,...xn} for X.

    since we have an isomorphism f, show that {f(x1),f(x2),...,f(xn)} is a basis for Y. we know that it has the same cardinality, because f is bijective,

    so linear independence is the only thing to prove.

    conversely, if X,Y have the same dimension, you need to show that a map which maps a basis {x1,x2,...,xn} for X bijectively to a basis {y1,y2,...,yn} for Y. can be extended to

    an isomorphism of X with Y. this map must be surjective (a consequence of the {yi} being a spanning set), injective (a consequence of the {xi} being linearly independent),

    and linear (which suggests that we extend our basis map linearly). even if you merely failed to specify that the {xi} and {yi} are bases, showing that f is bijective just on those sets,

    does not make f into a linear map on all of X. you need to say HOW you are defining f on an arbitrary element of X.
    So I just need to add let x_1,\cdots , x_n be a basis of X and y_1,\cdots , y_m be a basis for Y for part 1 and since it is an isomorphism we will get m=n.

    Let x_1,\cdots , x_n be a basis of X and y_1,\cdots , y_n be a basis for Y for part 2

    Yes?
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  4. #4
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    Re: Isomorphism

    um, no. you need to prove that m = n. one way to do that, is to DEFINE yi = f(xi). since f is bijective, this shows m = n, but it does not yet show that the {f(xi)} are actually a basis. perhaps they are linearly dependent...but this can't happen because...why? perhaps they don't span all of Y....but this can't happen...why?

    and again, just saying let {x1,x2,...,xn} be a basis for X, and let {y1,y2,..,yn} be a basis for Y, does not fix the problem. what is the isomorphism f for any particular (arbitrary) vector x in X?

    until you say what it is, how can you even begin to assert it is a linear map (vector space homomorphism)?
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  5. #5
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    Re: Isomorphism

    Quote Originally Posted by Deveno View Post
    um, no. you need to prove that m = n. one way to do that, is to DEFINE yi = f(xi). since f is bijective, this shows m = n, but it does not yet show that the {f(xi)} are actually a basis. perhaps they are linearly dependent...but this can't happen because...why? perhaps they don't span all of Y....but this can't happen...why?

    and again, just saying let {x1,x2,...,xn} be a basis for X, and let {y1,y2,..,yn} be a basis for Y, does not fix the problem. what is the isomorphism f for any particular (arbitrary) vector x in X?

    until you say what it is, how can you even begin to assert it is a linear map (vector space homomorphism)?
    I don't understand what you are trying to get at.
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  6. #6
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    Re: Isomorphism

    a map from basis to basis is NOT a map from all of X into Y.

    what is f(x) for an arbitrary x in X?
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