It seems in modern education, the link between linear algebra and complex algebra (I don't want to say "complex analysis", since the link is less apparent when you start talking about Cauchy integrals and whathaveyou) is most exemplified in an abstract algebra class, which is a shame! One learns that R is a field. Cool great! Then one learns the magic that C is just the adjunction of i=sqrt(-1) to R: C ~= R(i) ~= R[i] (~= meaning isomorophic). Then one is asked the question "prove there is no field which is a subfield of C and a field extension of R", which is extremely easy when you view field extensions as a vector space over the base field: [C\R] = 2 because C can be viewed as a vector space over R with bases {1, i}. Unfortunately the linear algebra usually stops there. On Sat, Mar 5, 2011 at 5:13 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Complex number algebra & vector algebra are usually taught as separate subjects, with little cross-over.
However, I find the following operations to be extremely useful -- at least in programming & Maxima hacking -- when dealing with complex numbers. They are often more useful/perspicuous than the conjugate operation.
dot(A,B) = dot(B,A) = realpart(A)*realpart(B)+imagpart(B)*imagpart(B) = (conjugate(A)*B+conjugate(B)*A)/2 = A.B
cross(A,B) = -cross(B,A) = determinant([realpart(A),imagpart(A)],[realpart(B),imagpart(B)]) = realpart(A)*imagpart(B)-imagpart(A)*realpart(B) = (conjugate(A)*B-conjugate(B)*A)/2i = AxB
1.A = dot(1,A) = realpart(A) i.A = dot(i,A) = imagpart(A)
1xA = cross(1,A) = imagpart(A) Axi = cross(A,i) = realpart(A)
(A.B)^2+(AxB)^2 = (A.A)*(B.B)
For example, the vector A can be decomposed into the parallel & perpendicular components of B:
A = (B.A)/(B.B)*B + i*(BxA)/(B.B)*B
Trying to express this just with conjugates isn't particularly perspicuous.
How come these vector operations on complex numbers aren't defined & used more often?
(So what if complex numbers are only 2-dimensional; they're incredibly useful in that 2D domain.)
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