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Math Help - Vectors in C^n

  1. #1
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    Vectors in C^n

    Hello:

    I have just completed an introductory Linear Algebra course at university, and I have a question. Is it possible to graph a vector in C^n if, perhaps, n is small e.g. n=2 or 3? I can't linearly order the complex numbers, but I can impose a lexicographic order upon them. However, this order precludes placing them on an axis because they become "infinitely dense".

    I'm still new to mathematics, and I have not studied complex analysis at all. I worked out of David Lay's "Linear Algebra and Its Applications" which I thought was a decent but not stellar text. Indeed, I had hoped that my text would go into real detail and abstraction. Alas, it did not.
    Last edited by Bilbo Baggins; December 31st 2009 at 01:23 PM.
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  2. #2
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    if n=2 , dim(C^2)=3 , you can draw in 3-dimensions space,
    if n>2, you only draw projection to 3-dimensions space.

    Current coordination couldn't support higher dimensions.
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  3. #3
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    Quote Originally Posted by math2009 View Post
    if n=2 , dim(C^2)=3 , you can draw in 3-dimensions space,
    if n>2, you only draw projection to 3-dimensions space.

    Current coordination couldn't support higher dimensions.
    Hi,

    Again, I know nothing of more advanced algebra or complex analysis. How would I graph a complex vector exactly? For example, <1+2i, 3+i> Where would this lie? You said it would have three dimensions, right? Then it's isomorphic to R^3? Thanks.
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  4. #4
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    Quote Originally Posted by math2009 View Post
    if n=2 , dim(C^2)=3 , you can draw in 3-dimensions space,
    How did you get that? As a vector space over the complex numbers, dim(C^2)= 2, of course, but as a vector space over the real numbers, dim(C^2)= 4, not 3.

    if n>2, you only draw projection to 3-dimensions space.

    Current coordination couldn't support higher dimensions.
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  5. #5
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    Quote Originally Posted by Bilbo Baggins View Post
    Hello:

    I have just completed an introductory Linear Algebra course at university, and I have a question. Is it possible to graph a vector in C^n if, perhaps, n is small e.g. n=2 or 3? I can't linearly order the complex numbers, but I can impose a lexicographic order upon them. However, this order precludes placing them on an axis because they become "infinitely dense".

    I'm still new to mathematics, and I have not studied complex analysis at all. I worked out of David Lay's "Linear Algebra and Its Applications" which I thought was a decent but not stellar text. Indeed, I had hoped that my text would go into real detail and abstraction. Alas, it did not.
    Actually, I came up with a bijection between C^n and R^2n that preserves vector addition and scalar multiplication as long as I restrict the scalar factors to the reals. So I suppose that the two are isomorphic. This leads to an interesting result. Given any basis for C^n, I can't span R^2n (and therefore C^n) with the images of that basis. I'm always n vectors short in R^2n.
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  6. #6
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    Quote Originally Posted by Bilbo Baggins View Post
    Actually, I came up with a bijection between C^n and R^2n that preserves vector addition and scalar multiplication as long as I restrict the scalar factors to the reals. So I suppose that the two are isomorphic. This leads to an interesting result. Given any basis for C^n, I can't span R^2n (and therefore C^n) with the images of that basis. I'm always n vectors short in R^2n.
    Yes, you can. Each a+ bi in C is mapped into (a, b) in R^2.
    Last edited by HallsofIvy; January 4th 2010 at 04:53 AM.
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