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Linear Algebra & Complex Geometry

  1. Jul 16, 2012 #1
    Hi I am new to the forum and I'm looking for help for a math problem I was given in my linear algebra class....

    Here its goes:

    Geometrically, a 1-dimensional subspace of C^2 is what kind of object? De- scribe the intersection of two such subspaces. Translate these geometric no- tions into algebraic expressions, and show algebraically that your geometric description was correct.


    -------

    So the first part is easy. A 1 dimensional subspace of C^2 is a line. but I'm confused as to what they are looking for for in this part "De- scribe the intersection of two such subspaces. Translate these geometric no- tions into algebraic expressions, and show algebraically that your geometric description was correct."


    Can someone please help! thanks!
     
  2. jcsd
  3. Jul 16, 2012 #2

    micromass

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    What is the intersection of two lines??
     
  4. Jul 16, 2012 #3
    Yeah I'm a little confused as to what kind of answer they are looking for.... that's why i'm here
     
  5. Jul 16, 2012 #4

    micromass

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    Can the intersection of two lines be a plane?? A line?? A point?? Empty??
     
  6. Jul 16, 2012 #5
    I mean it has to be the intersection of two 1-dimensional subspaces of C^2... so can't be a plane right ?

    I'm guessing a point. But I don't see how you can translate the intersection of two lines algebraically and proove that the description is correct....
     
  7. Jul 16, 2012 #6

    micromass

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    Yes, a point is correct. Can you tell which point??

    Also notice that the two lines can coincide, in that case the intersection will be a line.

    To see this algebraically, can you give the equation of a line in [itex]\mathbb{C}^2[/itex]?
     
  8. Jul 16, 2012 #7
    That is true. it could be two lines overlapping.

    In the case the the two lines are not overlapping. The intersection of two lines in C^2 must happen at the origin correct?

    so let's assume we have these two equations for the lines.

    Z1= 1 + i

    and

    Z2 = -1 + i

    They both intersect at the origin.

    Now what should I do next ?
     
  9. Jul 16, 2012 #8

    micromass

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    Right.

    Huh, those are not equations of lines. What do you mean with the above??
     
  10. Jul 16, 2012 #9
    Oops wait I think I'm mixing with the Reals here. K tell me if I'm right please.

    the equation of a line in a complex plane is of the form:

    (xo,yo) + t(u,v) where (xo,yo) is a point and (u,v) a vector ?

    So for you example the first line would have (0,0) as a point and we can assign an vector (1,1).

    And for the second line have the same starting point of (0,0) and a vector of (-1,1)?
     
  11. Jul 16, 2012 #10

    micromass

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    I don't see where you get the vectors (1,1) and (-1,1) from...

    The first line will have as equation

    [tex](x,y)=t(u,v)[/tex]

    and the second line will have

    [tex](x,y)=s(u^\prime,v^\prime)[/tex]

    Now, which points lie on both lines?
     
  12. Jul 16, 2012 #11
    well (x,y).

    Could you please explain why the structure of a line in a complex plain is of the the format

    (x,y) = t (u,v)

    please ?
     
  13. Jul 16, 2012 #12
    so are we looking to solve:

    s(u',v') = t(u,v) ?
     
  14. Jul 16, 2012 #13

    micromass

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    Yes, solve that for s and t.
     
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