## Monthly Archives: February 2013

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.

### 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.
$X+Y+XY$?