Lecture notes for Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 2005)

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

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

Ceci n'est pas un Math 55a syllabus (PS [PostScript] 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 may view them and print out copies in advance from the PostScript or PDF files linked below.

Metric Topology I (PS, PDF)
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)
Open and closed sets, and related notions
corrected 22.ix.05 (p.2: indices changed from i to alpha in Thm. 2.24 parts a,b;
p.4: closure of Br(p) vs. closed ball of radius r, also fixed typo ``subet'' for subset)

Metric Topology III (PS, PDF)
Introduction to functions and continuity
corrected 25.ix.05 (V for Z five times in p.3, paragraph 2)

Metric Topology IV (PS, PDF)
Sequences and convergence, etc.

Metric Topology V (PS, PDF)
Compactness and sequential compactness
corrected 1.x.05 to fix various minor typos

Metric Topology VI (PS, PDF)
Cauchy sequences and related notions (completeness, completions, and a third formulation of compactness)

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 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?

More will be said about exterior algebra when differential forms appear in Math 55b.

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

First & second problem set: Metric topology (PS, PDF)
corrected 19.ix.05 (clarification in 4(i), typos in 4(iv) and 11, parenthetical addendum in 11)
corrected again 20.ix.05: #6(ii*) had the wrong condition! :-( Shrenik Shah was the first to show it was impossible as first stated.

Third problem set: Metrics, sequences, compactness, and a bit of general topology (PS, PDF)

Fourth problem set: Topology grand finale (PS, PDF)

Fifth problem set / Linear Algebra I: vector space basics (PS, PDF)

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

Seventh problem set / Linear Algebra III: linear maps and duality (PS, PDF)

Eighth problem set / Linear Algebra IV: Eigenstuff, and a projective overture (PS, PDF)
corrected 9.xi.05 (typo in 1/iii)

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

Tenth problem set / Linear Algebra VI: Inner products, lattices, and normal operators (PS, PDF)
Here's my solution for problem 4 (and 3) (PS, PDF)

Eleventh problem set / Linear Algebra VII: Fourier foretaste, and symplectic structures (PS, PDF)

Twelfth and last problem set / Linear Algebra VIII: Groups, exterior algebra, and determinants (PS, PDF)
corrected 15.xii.05 (vi, not ||vi||, in 6(i); typo in 7 -- thanks to Scott K. for both;
also in 7, a sentence to say explicitly that k is fixed throughout;
and in the introductory paragraph for 8, the pairing is on V*, not V, and the map is from V* to V, not V to V*);
and (16.xii.05) again in 8, the ``associated pairing'' is now specified.