These are by Ian Le, the Fall 2003 Math 55a Course Assistant

If two real matrices A, B are conjugate (C^(-1)AC=B for some C) by a matrix (possibly complex) C, are they necessarily conjugate by a real matrix D?

What are the continuous functions from R to Q? from Q to R?

If we have a perfect pairing on a vector space V of dimension n, show that there does not exist a subspace of dimension greater than n/2 such that the pairing vanishes when restricted to this subspace.

What is the generalization of Taylor's theorem in more than one dimension?
What happens when you apply it to a polynomial in several variables?

Show that Abel summation can be derived from the product rule and the Stieltjes integral using a step function.

True or False?
In a compact metric space, every sequence converges.
The primary decomposition of a linear transformation is unique.
The Jordan canonical form of a linear transformation is unique.
The Jordan basis is unique.
There exists a function that is everywhere continuous but nowhere differentiable.
There exists a function with a jump discontinuity at every irrational number.
There exists a function with a jump discontinuity at exactly every rational number.
The limit of a convergent sequence of equicontinuous functions is equicontinuous.
If a sequence of differentiable functions converges uniformly to a limit, then the derivatives also converge.

Some problems:

1) Show that the series a_0+a_1+a_2+... converges iff the sum

sum_0^{infty} a_n/(sqrt{a_{n+1}+a_{n+2}+...})

converges.

2) Kuratowski's axioms state that on a set X there exists a closure operator "^c" such that the following hold for A, B subsets:

i) (null set)^c = null set
ii) A is containd in A^c
iii) (A^c)^c = A^c
iv) (A U B)^c = A^c U B^c

Show that this defines the topology of a space X, i.e., if we let the open sets of X be sets of the form X-A^c for any subset A of X, we obtain a topology for X.
Also show how to construct the operator ^c given a topology on X.

3) Let "_^_" denote the wedge of vector spaces/vectors. If S, T are linear transformations of V and W, respectively, what is det(S_^_T) in terms of det(S) and det(T)?
Let ST denote the symmetric product of S and T (the space of symmetric linear functionals from S* tensor T* to the ground field). What is det(ST)?
What are the eigenvalues and eigenvectors of S_^_T? ST?
What are their minimal polynomials?
Characteristic polynomials?
Jordan forms?

4) Let s be some permutation of N, the natural numbers.
Does there exist s such that the sum a_s(1)+a_s(2)+... converges whenever a_1+a_2+... converges, yet the opposite is not true?

5) Show that every real symmetric matrix A has a well defined cube root,i.e., a matrix B such that B^3=A. Can you show that B is unique among real matrices?