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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.

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 **R**^{n}
and other product spaces; isometries; boundedness and function spaces

IfSis an infinite set andXis an unbounded metric space then we can't use our definition ofX^{S}as a metric space because sup_{S}d(_{X}f(s),g(s)) might be infinite. But thebounded functionsfromStoXdoconstitute a metric space under the same definition ofd. A function is said to be ``bounded'' if its image is a bounded set. You should check that that_{XS}d(_{XS}f,g) is in fact finite for boundedfandg.

The ``Proposition'' on page 3 of the first topology handout can be extended as follows:Metric Topology II (PS, PDF, PDF')iv) ForIn other words, for everyeverypointpofXthere exists a real numberMsuch thatd(p,q)<Mfor allqofE.pinXthere exists an open ball aboutpthat containsE. Do you see why this is equivalent to (i), (ii), and (iii)?

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 *N _{r}* to

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 integratingx.^{t}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:

- [cf. Axler, p.3]
Unless noted otherwise,
**F**may be an arbitrary field, not only**R**or**C**. The most important fields other than those of real and complex numbers are the field**Q**of rational numbers, and the finite fields**Z**/*p***Z**(*p*prime). Other examples are the field**Q**(*i*) of complex numbers with rational real and imaginary parts; more generally,**Q**(*d*^{1/2}) for any nonsquare rational number*d*; the ``*p*-adic numbers''**Q**_{p}(*p*prime), introduced at the end of our topology unit; and more exotic finite fields such as the 9-element field (**Z**/*3***Z**)(*i*). Here's a review of the axioms for fields, vector spaces, and related mathematical structures. - [cf. Axler, p.22] We define the span of an arbitrary subset
*S*of (or tuple in) a vector space*V*as follows: it is the set of all (finite) linear combinations*a*_{1}*v*_{1}+ ... +*a*with each_{n}v_{n}*v*in_{i}*V*and each*a*in_{i}*F*. This is still the smallest vector subspace of*V*containing*S*. In particular, if*S*is empty, its span is by definition {0}. We do*not*require that*S*be finite. - Unlike Axler, we spend some time on ``quotient vector spaces'', described a bit more fully in the Q & A page.
- Axler also unaccountably soft-pedals the important notion of
**duality**. - Here's a brief preview of abstract nonsense (a.k.a. diagram-chasing), and a diagram-chasing interpretation of quotients and duality.
- Axler proves the Fundamental Theorem of Algebra using complex analysis, which cannot be assumed in Math 55. Here's a proof using the topological tools we developed in the first month of class, in PS, PDF, and PDF'. (Axler gives the complex-analytic proof on page 67.)
- We shall need some ``eigenstuff'' also in an infinite-dimensional setting, so will not assume that any vector space is (nonzero) finite dimensional unless we really must.
- If
*T*is a linear operator on a vector space*V*, and*U*is an invariant subspace, then the quotient space*V*/*U*inherits an action of*T*. Moreover, the annihilator of*U*in*V*^{*}is an invariant subspace for the action of the adjoint operator*T*^{*}on*V*^{*}. (Make sure you understand why both these claims hold.) - Triangular matrices are intimately related with ``flags''.
A
*flag*in a finite dimensional vector space*V*is a sequence of subspaces {0}=*V*_{0},*V*_{1},*V*_{2}, ...,*V*, with each_{n}=V*V*of dimension_{i}*i*and containing*V*_{i-1}. A basis*v*_{1},*v*_{2}, ...,*v*determines a flag:_{n}*V*is the span of the first_{i}*i*basis vectors. Another basis*w*_{1},*w*_{2}, ...,*w*determines the same flag if and only if each_{n}*w*is a linear combination of_{i}*v*_{1},*v*_{2}, ...,*v*(necessarily with nonzero_{i}*v*coefficient). The_{i}*standard flag*in*F*is the flag obtained in this way from the standard basis of unit vectors_{n}*e*_{1},*e*_{2}, ...,*e*. The punchline is that, just as a diagonal matrix is one that respects the standard basis (equivalently, the associated decomposition of_{n}*V*as a direct sum of 1-dimensional subspaces),*an upper-triangular matrix is one that respects the standard flag.*Note that the*i*-th diagonal entry of a triangular matrix gives the action on the one-dimensional quotient space*V*/_{i}*V*_{i-1}(each*i*=1,...,*n*).

- One of many applications is the
**trace**of an operator on a finite dimensional*F*-vector space*V*. This is a linear map from Hom(*V*,*V*) to*F*. We can define it simply as the composition of two maps: our identification of Hom(*V*,*V*) with the tensor product of*V*^{*}and*V*, and the natural map from this tensor product to*F*coming from the bilinear map taking (*v*^{*},*v*) to*v*^{*}(*v*). - Here are some basic facts about general
**norms**on real and complex vector spaces.**(revised 21.xi.02)** - The key trick in proving ``Sylvester's Law of Inertia'' is this:
Suppose the finite-dimensional space
*V*over**R**(or**C**) is the orthogonal direct sum of subspaces*U*_{1},*U*_{2}under a bilinear symmetric pairing (or a sesquilinear conjugate-symmetric pairing) that is positive definite on*U*_{1}and negative definite on*U*_{2}. Then any subspace*W*of*V*on which the pairing is positive definite has dimension no greater than dim(*U*_{1}). Proof: On the intersection of*W*with*U*_{2}, the pairing is both positive and negative definite; hence that subspace is {0}. The claim follows by a dimension count. - Over any field not of characteristic 2,
one can show in much the same way that for any non-degenerate
symmetric pairing on a finite-dimensional vector space
there is an orthogonal basis, or equivalently
a choice of basis such that the pairing is
(
*x*,*y*)=sum_{i}(*a*) for some nonzero scalars_{i}x_{i}y_{i}*a*. But in general it can be quite hard to decide whether two different collections of_{i}*a*yield isomorphic pairings. Even over_{i}**Q**the answer is already tricky in dimensions 2 and 3, and I don't think it's known in a vector space of arbitrary dimension. - Here's an online version of the notes for the lecture on lattices.
- All of Chapter 8 works over an arbitrary algebraically closed field,
not only over
**C**; and the first section (``Generalized Eigenvalues'') works over any field. - We don't stop at Corollary 8.8: let
*T*be any operator on a vector space*V*over a field*F*, not assumed algebraically closed. If*V*is finite-dimensional, then The Following Are Equivalent:

(1) There exists a nonnegative integer*k*such that*T*=0;^{k}

(2) For any vector*v*, there exists a nonnegative integer*k*such that*T*=0;^{k}v

(3)*T*=0, where^{n}*n*=dim(*V*).

Note that (1) and (2) make no mention of the dimension, but can still fail for operators on infinite-dimensional spaces. We readily deduce the further equivalent conditions:

(4) There exists a basis for*V*for which*T*has an upper-triangular matrix with every diagonal entry equal zero;

(5) Every upper-triangular matrix for*V*has zeros on the diagonal. - The space of generalized 0-eigenvalues
(the maximal subspace on which
*T*is nilpotent) is sometimes called the*nilspace*of*T*. It is an invariant subspace. When*V*is finite dimensional,*V*is the direct sum of the nilspace and another invariant subspace*V'*, consisting of the intersection of the subspaces*T*(^{k}*V*) as*k*ranges over all positive integers. This can be used to quickly prove Theorem 8.23 and consequences such as Cayley-Hamilton (Theorem 8.20). - The dimension of the space of generalized
*c*-eigenvalues (i.e., of the nilspace of*T-cI*) is usually called the*algebraic*multiplicity of*c*(since it's the multiplicity of*c*as a root of the characteristic polynomial of*T*), to distinguish it from the ``geometric multiplicity'' which is the dimension of ker(*T-cI*).

Some more tidbits about exterior algebra:

- If
*w*,*w*' are elements of the*m*-th and*m*'-th exterior powers of*V*, then*ww*'=(-1)^{mm'}*w*'*w*; that is,*w*and*w*' commute unless*m*and*m*' are both odd in which case they anticommute. - If
*m*+*m*'=*n*=dim(*V*) then the natural pairing from the*m*-th and*m*'-th exterior powers to the*n*-th is nondegenerate, and so identifies the*m*'-th exterior power canonically with the dual of the*m*-th*tensored with the top (n-th) exterior power*. - In particular, if
*m*=1, and*T*is any invertible operator on*V*, then we find that the induced action of*T*on the (*n*-1)st exterior power is the same as its action on*V*^{*}multiplied by det(*T*). This yields the formula connecting the inverse and cofactor matrix of an invertible matrix (a formula which you may also know in the guise of ``Cramer's rule''). One consequence of this formula is that an*n*-by-*n*integer matrix*A*had an inverse with integer entries if and only if |det(*A*)|=1. This in turn implies that |det(*M*)| is an invariant of the lattice*L*=*M***Z**^{n}. In 55b we'll interpret this invariant as the ``covolume'' of*L*, that is, the volume of the*n*-dimensional torus**R**^{n}/*L*. - For each
*m*there is a natural non-degenerate pairing between the*m*-th exterior powers of*V*and*V*^{*}, which identifies these exterior powers with each other's dual.

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')

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 existS,Tsuch thatST-TS=Iif and only ifnis a multiple of the characteristic ofF(whether this characteristic is zero or not). This generalizes Corollary 10.13 in Axler; recall that Axler requires that F=RorC, 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 subspaceEleventh and last problem set / Linear Algebra VII: Fourier foretaste, determinants, and a grand pfinale (PS, PDF, PDF')WwithW^{\perp}={0} is the space of functions whose integral from 0 to 1/2 vanishes. ThisWis closed because the functional takingfto the integral offover [0,1/2] is continuous (with delta=sqrt(2)*epsilon). In the completion ofV, 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 inVitself.

Concerning Problem 4: we'll obtain a much more precise result in Math 55b by applying Fourier analysis on toriR^{n}/L(La lattice inR^{n}).

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!