Linear Algebra & Complex Geometry

In summary, the conversation involves discussing the geometric and algebraic representations of 1-dimensional subspaces of C^2. The main question is about the intersection of two such subspaces, which is determined to be a point. The equations of lines in a complex plane are also discussed, and the conversation ends with solving for s and t in the equation s(u',v') = t(u,v).
  • #1
Wlaubz
7
0
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!
 
Physics news on Phys.org
  • #2
What is the intersection of two lines??
 
  • #3
Yeah I'm a little confused as to what kind of answer they are looking for... that's why I'm here
 
  • #4
Can the intersection of two lines be a plane?? A line?? A point?? Empty??
 
  • #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...
 
  • #6
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
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
Wlaubz said:
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
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
Wlaubz said:
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
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
so are we looking to solve:

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

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

Yes, solve that for s and t.
 

1. What is the definition of linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations, matrices, and vector spaces. It involves the use of algebraic operations to manipulate and solve systems of linear equations.

2. What is the importance of linear algebra in scientific fields?

Linear algebra has many applications in scientific fields such as physics, engineering, and computer science. It is used to model and solve complex systems, analyze data, and make predictions. It is also the foundation for advanced mathematical concepts like calculus and differential equations.

3. What are the key concepts in linear algebra?

The key concepts in linear algebra include vectors, matrices, determinants, eigenvalues and eigenvectors, and linear transformations. These concepts are used to solve systems of equations, perform matrix operations, and analyze the properties of geometric objects.

4. How does linear algebra relate to complex geometry?

Linear algebra and complex geometry are closely related as they both deal with geometric objects and mathematical operations. Linear algebra provides the tools and techniques to study and manipulate these objects, while complex geometry uses these tools to understand the properties and behavior of complex geometric shapes.

5. What are some real-world applications of linear algebra and complex geometry?

Some real-world applications of linear algebra and complex geometry include computer graphics, image and signal processing, machine learning, and quantum mechanics. They are also used in fields such as economics, social sciences, and statistics to model and analyze complex systems and data sets.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
407
  • Calculus and Beyond Homework Help
Replies
10
Views
937
  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
5K
  • Calculus and Beyond Homework Help
Replies
14
Views
398
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
534
  • Calculus and Beyond Homework Help
Replies
25
Views
2K
Back
Top