Recent content by morphism

You should look in a linear book (or online) to see how one typically finds eigenvectors. And while you're at it, look up "diagonalization". The point of diagonalization here is this: if D is diagonal, say with entries d_1, ..., d_n down the diagonal, then e^D will be diagonal entries...

What would g be in this case?

Sounds like you've pretty much got it. Care to summarize your argument in one post, so we can make sure there are no gaps?

There's no reason to be ashamed. I only wanted to know what flavor of the subject you enjoy before suggesting people/schools. Perhaps you can read up on the different types of representation theory and then let me know which ones strike your fancy. If you're not sure where to begin, I can make...

Each R will be what you get if you do the row operation to the identity matrix. E.g. if you do the row operation that swaps rows 1 and 3, then R will be the identity matrix with rows 1 and 3 swapped.

What if 0 is an eigenvalue? No. Consider ##\left( \begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix} \right)##. If A^2=-I then det(A^2)=det(-I). Try to see if you can use this to answer your question.

I think what's confusing you is the difference between an "internal direct sum" (in which you write a vector space as a direct sum of subspaces) and an "external direct sum" (in which you take two vector spaces and add them together, without reference to them being contained in a larger vector...

Typically that's referred to as "equivalence", but anyway, your idea does work, i.e. it will produce an equivalent normal series.

The rref of A is of the form GA for some invertible matrix G, so the solution sets to Ax=b and rref(A)x=b will generally be different. The precise fact to note is: if Ax=b then rref(A)x=(GA)x=G(Ax)=Gb. Note that if b=0 then the previous computation yields rref(A)x=0; and conversely, if...

Well done!

A polynomial of odd degree always has a real root, and the denominator can't have a root if we want the rational function to be defined on all of R.

Contradicts what fact? This is fine, but your write-up is sloppy.

Can you express the determinant of AB in terms of those of A and B?

What does it mean to say that two normal series are isomorphic?

A rational function is the ratio of two polynomials. You observed if there's anything "funky" in the denominator, then certainly R won't get mapped to (0,1). Now just think about what "funky" really means.. Remember, the denominator is a polynomial. Edit: Sorry--that's not really helpful...