Monthly Archives: February 2013

Quadratic forms

A quadratic form is essentially the same thing as a symmetric bilinear form, at least over a field of characteristic different from 2. A while ago, I wrote a brief note on quadratic forms dealing with diagonalization and geometry. I hope it’s useful for HT2013, sheet 4, problem 8, in particular.

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Orthogonal bases

I was a bit confused during tutorials today about orthogonal bases, so I wrote up a short note on how to find them. This method is illustrated with problem 5 on sheet 4 for Linear Algebra II.

Probability sheet 4

Here is a sketch of problem 10 on sheet 4. The convergence question that came up in the tutorial is clarified in an elementary way.

Added: Towards the end of the document, I should have stated that S_N is a Riemann sum for the integral with mesh size 2^{-N+1}, Since this goes to zero and we are integrating a continuous function, we get convergence of the sum to the integral.

Homomorphism of formal groups

Wei Yue asks the following question: In the definition of a homomorphism f(T) between two formal groups F and G, what is the reason we required f\in TR[[T]]? The short reason is that we want f to preserve the origin, that is, f(0)=0. A more complicated reason is that in general, the expression G(f(X), f(Y)) may not make sense for a power series G(X,Y) if f has a non-zero constant term. This is a point we have been somewhat cavalier about: When can we substitute one power series in another and get a well-defined power series as the answer? You should ponder this for yourself a bit, and then try to formulate some conditions precisely using the definition

R[[T]]= \lim_{\leftarrow} R[T]/(T^n)

of the ring of power series with coefficients in R.

Here is an exercise worth trying out: Suppose F(X,Y) is a polynomial that defines an algebraic group law on the field \mathbb{C} with 0 as the origin. Then F(X,Y)=X+Y is the only choice.
Hang on, what about

X+Y+XY?

Well, you have to formulate a bit carefully what is meant by an algebraic group law. Anyways, the conclusion is that deforming the usual group structure in any reasonable sense requires us to move out of the realm of polynomials.