Linear Algebra & Complex Geometry

[Wlaubz]

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.


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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!

 

[micromass]

What is the intersection of two lines??

 

[Wlaubz]

Yeah I'm a little confused as to what kind of answer they are looking for... that's why I'm here

 

[micromass]

Can the intersection of two lines be a plane?? A line?? A point?? Empty??

 

[Wlaubz]

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...

 

[micromass]

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 \mathbb{C}^2?

 

[Wlaubz]

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 ?

 

[micromass]

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??

 

[Wlaubz]

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)?

 

[micromass]

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

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

and the second line will have

(x,y)=s(u^\prime,v^\prime)

Now, which points lie on both lines?

 

[Wlaubz]

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 ?

 

[Wlaubz]

so are we looking to solve:

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

 

[micromass]

so are we looking to solve:

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

Yes, solve that for s and t.