Linear Algebra & Complex Geometry

  • Thread starter Wlaubz
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  • #1
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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!
 

Answers and Replies

  • #2
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What is the intersection of two lines??
 
  • #3
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Yeah I'm a little confused as to what kind of answer they are looking for.... that's why i'm here
 
  • #4
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Can the intersection of two lines be a plane?? A line?? A point?? Empty??
 
  • #5
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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....
 
  • #6
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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]?
 
  • #7
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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 ?
 
  • #8
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3,297
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?

Right.

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

Z1= 1 + i

and

Z2 = -1 + i

Huh, those are not equations of lines. What do you mean with the above??
 
  • #9
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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)?
 
  • #10
22,129
3,297
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)?

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?
 
  • #11
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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 ?
 
  • #12
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so are we looking to solve:

s(u',v') = t(u,v) ?
 
  • #13
22,129
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so are we looking to solve:

s(u',v') = t(u,v) ?

Yes, solve that for s and t.
 

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