Lecture notes for Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 2002)

If you find a mistake, omission, etc., please let me know by e-mail.

Andrei's Math 55 page
Q & A: Questions that arose concerning lectures, problem sets, etc., and my replies

The orange balls mark our current location in the course, and the current problem set.

Ceci n'est pas un Math 55a syllabus (PS or PDF or PDF')

Our first topic is the topology of metric spaces, a fundamental tool of modern mathematics that we shall use mainly as a key ingredient in our rigorous development of differential and integral calculus. To supplement the treatment in Rudin's textbook, I wrote up 20-odd pages of notes in six sections; copies will be distributed in class, and you also view them and print out copies in advance from the PostScript or PDF files linked below.

Metric Topology I (PS, PDF, PDF') corrected 24.ix.02 (see top of page 2)
Basic definitions and examples: the metric spaces Rn and other product spaces; isometries; boundedness and function spaces

If S is an infinite set and X is an unbounded metric space then we can't use our definition of XS as a metric space because supS dX(f(s),g(s)) might be infinite. But the bounded functions from S to X do constitute a metric space under the same definition of dXS. A function is said to be ``bounded'' if its image is a bounded set. You should check that that dXS(f,g) is in fact finite for bounded f and g.
The ``Proposition'' on page 3 of the first topology handout can be extended as follows:
iv) For every point p of X there exists a real number M such that d(p,q)<M for all q of E.
In other words, for every p in X there exists an open ball about p that contains E. Do you see why this is equivalent to (i), (ii), and (iii)?
Metric Topology II (PS, PDF, PDF')
Open and closed sets and related notions

Metric Topology III (PS, PDF, PDF') corrected 30.ix.02 (see top of page 2)
Introduction to functions and continuity

Metric Topology IV (PS, PDF, PDF') corrected 30.ix.02 (bottom of page 2: continuity [not ``convergence''] of the functions...)
Sequences and convergence, etc.
(several more typos corrected 2.x.02)

Metric Topology V (PS, PDF, PDF') corrected 3.x.02, mostly to fix typos and change Nr to Br
Compactness and sequential compactness

Metric Topology VI (PS, PDF, PDF') updated 7.x.02, mainly to mention diagonal subsequences
Cauchy sequences and related notions (completeness, completions, and a third formulation of compactness)
corrected 11.x.02: a continuous real-valued function on a nonempty compact space attains... (page 2)

Here's Fermat's trick for integrating xt dx.

at least in the beginning of the linear algebra unit, we'll be following the Axler textbook closely enough that supplementary lecture notes should not be needed. Some important extensions/modifications to the treatment in Axler:

Less surprising than the absence of quotients and duality in Axler is the lack of tensor algebra. That won't stop us in Math 55, though. Here's an introduction in PS, PDF, and PDF'. [As you might guess from \oplus, the TeXism for the tensor-product symbol is \otimes.] We'll define the determinant of an operator T on a finite dimensional space V as follows: T induces a linear operator T' on the top exterior power of V; this exterior power is one-dimensional, so an operator on it is multiplication by some scalar; det(T) is by definition the scalar corresponding to T'. The ``top exterior power'' is a subspace of the ``exterior algebra'' of V, which is the quotient of the tensor algebra by the ideal generated by {v*v: v in V}. We'll still have to construct the sign homomorphism from the symmetry group of order dim(V) to {1,-1} to make sure that this exterior algebra is as large as we expect it to be, and that in particular that the (dim(V))-th exterior power has dimension 1 rather than zero.

Some more tidbits about exterior algebra:

We'll also show that a symmetric (or Hermitian) matrix is positive definite iff all its eigenvalues are positive iff it has positive principal minors (the ``principal minors'' are the determinants of the square submatrices of all orders containing the (1,1) entry). More generally we'll show that the eigenvalue signs determine the signature, as does the sequence of signs of principal minors. For positive definiteness, we have the two further equivalent conditions: the symmetric (or Hermitian) matrix A=(aij) is positive definite iff there is a basis (vi) of Fn such that aij=<vi,vj> for all i,j, and iff there is an invertible matrix B such that A=BB*. For example, the matrix with entries 1/(i+j-1) (``Hilbert matrix'') is positive-definite, because it is the matrix of inner products (integrals on [0,1]) of the basis 1,x,x2,...,xn-1 for the polynomials of degree <n. Can you find the determinant of this matrix?

We shall say more about exterior algebra when we discuss differential forms in Math 55b.

Here's a brief introduction to field algebra and Galois theory.

Here's a batch of practice/review problems for the material covered in Math 55a (PS, PDF, PDF').

First problem set: Metric topology (PS, PDF, PDF') corrected 24.ix.02 (first display of page 2)
Andrei's solution set (PS, PDF, PDF')

Second problem set: Metrics, topologies, continuity, and sequences (PS, PDF, PDF') corrected 29.ix.02 (see problem 5)

Third problem set: Sequences, function spaces, and compactness (PS, PDF, PDF')
Andrei's solution set (PS, PDF, PDF')

Problems 4 and 8 are theorems of Urysohn and Lebesgue respectively (the latter usually known as the ``Lebesgue Covering Lemma'').
Fourth problem set: Topology grand finale (PS, PDF, PDF')
Problem 4 is the Arzela-Ascoli Theorem.
Fifth problem set / Linear Algebra I: vector space basics (PS, PDF, PDF')

Sixth problem set / Linear Algebra II: the dimension and some of its uses (PS, PDF, PDF')

Seventh problem set / Linear Algebra III: linear maps and duality (PS, PDF, PDF') corrected 5.xi.02 (x's instead of a's in problem 8)

Eighth problem set / Linear Algebra IV: Eigenstuff (PS, PDF, PDF')

Ninth problem set / Linear Algebra V: Tensors, etc. (PS, PDF, PDF')

For Problem 2: there exist S,T such that ST-TS=I if and only if n is a multiple of the characteristic of F (whether this characteristic is zero or not). This generalizes Corollary 10.13 in Axler; recall that Axler requires that F=R or C, both of which have characteristic zero.

Tenth problem set / Linear Algebra VI: Inner products, lattices, and normal operators (PS, PDF, PDF')

In problem 2, an example of a proper closed subspace W with W\perp={0} is the space of functions whose integral from 0 to 1/2 vanishes. This W is closed because the functional taking f to the integral of f over [0,1/2] is continuous (with delta=sqrt(2)*epsilon). In the completion of V, this subspace is the orthogonal complement of the 1-dimensional space generated by the characteristic function of the interval [0,1/2]; but this characteristic function is not in V itself.
Concerning Problem 4: we'll obtain a much more precise result in Math 55b by applying Fourier analysis on tori Rn/L (L a lattice in Rn).
Eleventh and last problem set / Linear Algebra VII: Fourier foretaste, determinants, and a grand pfinale (PS, PDF, PDF') corrected 16.xii.02 [missing factor of n! in the Pfaffian pformula :-( ]

Concerning Problem 4: the order-4 Rubik's Cube allows for hidden transpositions of edge ``cubies'' (which come in indistinguishable pairs), making it possible to fake a simple transposition by composing it with a hidden switch to create an even permutation!