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Planes in C^nby WWGD
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#1
May612, 06:23 PM

P: 529

Hi, All:
Just curious: What is the general equation of a linear nsubspace S of C^n (n>1)? I can figure out a line in C^n, but can't think of strict higherd subspaces S, i.e., S is not the whole space. Thanks for any refs, hints. 


#2
May612, 09:30 PM

P: 606

If you know the real spaces then you're done, as [itex]\,\,\mathbb{C}^n_\mathbb{R}\cong \mathbb{R}^{2n}_\mathbb{R}\,\,[/itex] ... Now, if you're talking of [itex]\,\,\mathbb{C}^n_\mathbb{C}\,\,[/itex] then this is "basically the same" as [itex]\,\,\mathbb{R}^n_\mathbb{R}\,\,[/itex]: a kdimensional subspace can be given as the solution to some system of linear equations. DonAntonio 


#3
May1012, 08:28 PM

P: 529

Thanks, Don Antonio, I thought something very similar:
We use the fact that linear maps take subspaces to subspaces, and linear isomorphisms actually preserve the dimension of these subspaces. I believe then, this isomorphism would transform the equation of an nsubspace in R^2n into one in C^n. 


#4
May1012, 08:59 PM

P: 529

Planes in C^n
1) Choose a basis B_S for S , and extend to a basis B_V for all of V. Choose a basis B_W for W 2)Define a linear map by defining: T(b_V)==0 , for all b_S in B_S 3)Set up a bijection between basis vectors in B_V\B_S and basis vectors in B_W. 4)Extend by linearity. 


#5
May3112, 07:45 PM

P: 529

Sorry, I forgot the original question I had:
Can we represent a line in C^n as a function of z ( a single complex variable)? If f: R>R is linear, we can represent it as f(x)=ax+b, i.e., as a function of a single real variable. Can the same be done for lines in C^n ? 


#6
Jun112, 07:13 AM

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P: 39,345

No, you cannot write a line, a one dimensional subspace, of C^{n} as a single equation in z. Saying that a subspace "has dimension i" means that you need to know i of its coordinates of a given point but can calculate the other n i. And that means you will need to have n i equations to calculate those.
This has nothing to do with "C" versus "R". Back in Calculus you saw that a single equation can determine a line in two dimension, but a single equation in R^{3} determines a plane, of dimension 2. In order to specify a line, of dimension 1, you need two equations. Geometrically, that can interpreted as the line of intersection of the two planes given by the two equations. Of course, you can specify a line in R^{3} by 3 parametric equations, giving x, y, and z as functions of the single variable, t. But that is now three equations in four variables and so still has 4 3= 1 degree of freedom. In general, to specify an i dimensional subspace of an n dimensional space, you need n i equations. 


#7
Jun112, 07:06 PM

P: 529

And ℂ^{2}, has complex dimension 2, and so we would need 21=1 , so what am I missing? 


#8
Jun212, 09:11 AM

P: 4,572

You could embed a line or a twodimensional plane (proper dimension, not dimension of embedded space) instead any higher dimensional space and the intrinsic dimension is still unchanged. 


#9
Jun312, 11:25 AM

P: 529

According to Ivy, to write an idim space in an ambient nspace, ni variables would be enough. But , if I understood well, this would lead to a line in ℂ^{1} requiring ni =11=0 equations and for a line embedded in ℂ^{2} needing ni=21 equations, which does not work, from previous posts. 


#10
Jun412, 01:51 AM

P: 4,572

If you want to describe a plane for ndimensional complex numbers, then simply define a linear object with your embedded space, and your number of intrinsic dimensions for your object (exactly as HallsOfIvy has pointed out) and then check that they obey the plane equation and also linearity. If you want to deal with projections and so on, define the appropriate projection that maintains the properties of a projection (P^2 = P) and make sure you get the other expected properties that you get for cartesiantype projections (assuming you are dealing with a flat plane). If are only dealing with the situation where are only dealing with addition and scalar multiplication of complex numbers, then you are done. If you wish to however deal with multiplication of complex numbers, then this will change things a lot, but multiplication is not really in a linear context: linear usually refers to having axioms f(X+Y) = f(X) + f(Y) and f(aX) = af(X): anything involving multiplication is different. 


#11
Jun412, 04:32 PM

P: 529

Thanks, Chiro, I was thinking along those lines. And, yes, my problem
is multiplication, because I am also considering the complexes as a vector space over C, which implies that for f(aX)=af(X) , both aX and af(X) are multiplications of complexes. And, I guess , re your 1st paragraph, that a line from C to C is not a function of a complex variable. My idea was to use the isomorphism between C as a V.S over R and R^2 also as a V.S over R, to map lines to lines, since linear isomorphisms not only send subspaces to subspaces, but also preserve the dimension of the subspaces. Ultimately, I am trying to get a better understanding of complex projective spaces, and for that I need to set up the lines before understanding what their projections look like. There seems to be a big difference between a line in C as a V.S over R, and a line in C as a V.S over itself. 


#12
Jun412, 05:31 PM

P: 529

I am trying to replicate the idea/method to describe how any subspace of a real vector
space is the kernel of a linear map*, to complex vector spaces. My confusion had to see with the difference between the identification z~λz (with z seen as a vector (x,y)), when λ is real, vs. when it is complex ( i.e., has nonzero imaginary part ). In the first case z~λz gives us a plane thru the origin ; we get all rotations and scalings of z, since multiplying by a complex number has the effect of rotating it and stretching it. In the last case, we get a line, since real multiples of z only scale but do not rotate z. *The method I know for having a subspace S of V described as the kernel of a map T:V>W (where W has to be largeenough ), is that of defining T(v_s)==0 for any v_s in the basis of S , extending that basis of S into a basis for V, and then setting up a bijection between vectors b_v in the extended basis into any basis for W (so we need W to be large enough dimension wise to be able to define that bijection. ) 


#13
Jun412, 10:04 PM

P: 4,572

Have you looked at the theory for complex vector spaces?



#14
Jun512, 07:40 PM

P: 529

in a standard, clear way. Can you suggest a good source? 


#15
Jun512, 09:06 PM

P: 4,572

Doing a quick google search (for complex vector spaces) gives this: http://math.stackexchange.com/questi...vectorspaces There are plenty of PDF's for the geometry of complex vector spaces which indirectly answers all your questions (look at the inner and outer products and related projections involved in complex vector spaces). 


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