Geometry of Complex Vector Spaces
Stereographic projection.
Let the coordinates in Rn+1 be u0 , ..., un . The locus
(1) u20 + · · · + u2n = 1
of unit length vectors is called an n-dimensional sphere, and is often denoted by S n . Its dimension n is the
number of degrees of freedom of a point on the locus. So S 2 is the usual unit sphere in 3-space, and S 1 is
the unit circle in the plane.
Stereographic projection is useful for visualizing the sphere, especially the 3-sphere S 3 . Via stereographic
projection, the points of the n-sphere correspond bijectively to points of the n-dimensional hyperplane H
defined by the equation u0 = 0. Though it is not traditional, I like to depict the first coordinate u0 as
the “vertical” axis. The north pole is the point (1, 0, . . . , 0) at the top of the sphere. The stereographic
projection of a point p = (u0 , ..., un ) on S n is the intersection of the line through the north pole and the
point p with H. This projection is bijective except at the north pole, where it is not defined. One says that
the north pole is sent to “infinity”.
In parametric form, the line of projection is (t(u0 − 1) + 1, tu1 , ..., tun ), and the intersection with H is the
point
u1 un
(2) (0, y1 , ..., yn ) = (0, , ..., ).
1 − u0 1 − u0
Writing r2 = y12 + · · · + yn2 , the inverse function sends the point (0, y1 , ..., yn ) to
r2 − 1 2y1 2yn
(3) (u0 , ..., un ) = ( 2
, 2 , ..., 2 ).
r +1 r +1 r +1
The complex vector space Cn .
Let V denote the complex vector space Cn . We may separate a complex vector X = (x1 , ..., xn )t into its real
and imaginary parts, writing xν = aν + bν i so that X = A + Bi, where A = (a1 , ..., an )t and B = (b1 , ..., bn )t .
In this way, the complex n-dimensional vector X corresponds to a pair of n-dimensional real vectors, or to a
single real vector of dimension 2n. In the long run it is better not to introduce a separate real vector, but let’s
do so for now, and denote this 2n-dimensional real vector by X. How the entries aν , bν of this 2n-dimensional
real vector are arranged is arbitrary. We’ll use the arrangement X = (a1 , b1 , a2 , b2 , ..., an , bn )t .
Thus there is a natural bijective correspondence between Cn and R2n . This bijection is not called an
isomorphism of vector spaces because the field of scalars in Cn is the field of complex numbers, while in R2n
it is the field of real numbers.
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Stereographic projection.
Let the coordinates in Rn+1 be u0 , ..., un . The locus
(1) u20 + · · · + u2n = 1
of unit length vectors is called an n-dimensional sphere, and is often denoted by S n . Its dimension n is the
number of degrees of freedom of a point on the locus. So S 2 is the usual unit sphere in 3-space, and S 1 is
the unit circle in the plane.
Stereographic projection is useful for visualizing the sphere, especially the 3-sphere S 3 . Via stereographic
projection, the points of the n-sphere correspond bijectively to points of the n-dimensional hyperplane H
defined by the equation u0 = 0. Though it is not traditional, I like to depict the first coordinate u0 as
the “vertical” axis. The north pole is the point (1, 0, . . . , 0) at the top of the sphere. The stereographic
projection of a point p = (u0 , ..., un ) on S n is the intersection of the line through the north pole and the
point p with H. This projection is bijective except at the north pole, where it is not defined. One says that
the north pole is sent to “infinity”.
In parametric form, the line of projection is (t(u0 − 1) + 1, tu1 , ..., tun ), and the intersection with H is the
point
u1 un
(2) (0, y1 , ..., yn ) = (0, , ..., ).
1 − u0 1 − u0
Writing r2 = y12 + · · · + yn2 , the inverse function sends the point (0, y1 , ..., yn ) to
r2 − 1 2y1 2yn
(3) (u0 , ..., un ) = ( 2
, 2 , ..., 2 ).
r +1 r +1 r +1
The complex vector space Cn .
Let V denote the complex vector space Cn . We may separate a complex vector X = (x1 , ..., xn )t into its real
and imaginary parts, writing xν = aν + bν i so that X = A + Bi, where A = (a1 , ..., an )t and B = (b1 , ..., bn )t .
In this way, the complex n-dimensional vector X corresponds to a pair of n-dimensional real vectors, or to a
single real vector of dimension 2n. In the long run it is better not to introduce a separate real vector, but let’s
do so for now, and denote this 2n-dimensional real vector by X. How the entries aν , bν of this 2n-dimensional
real vector are arranged is arbitrary. We’ll use the arrangement X = (a1 , b1 , a2 , b2 , ..., an , bn )t .
Thus there is a natural bijective correspondence between Cn and R2n . This bijection is not called an
isomorphism of vector spaces because the field of scalars in Cn is the field of complex numbers, while in R2n
it is the field of real numbers.
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