Lecture notes for
Math 55a: Honors Advanced Calculus and Linear Algebra
(Fall 1999)
If you find a mistake, omission, etc., please
let me know
by email.
The orange balls
mark our current location in the course,
and the current problem set.
h1.ps:
Ceci n'est pas un Math 55a syllabus.
top1.ps:
Metric Topology I, basic definitions and examples
(the metric spaces R^{n}
and other product spaces; isometries; boundedness and function
spaces)
top2.ps:
Metric Topology II, open and closed sets and related notions
top3.ps:
Metric Topology III, introduction to functions and continuity
top4.ps:
Metric Topology IV, sequences and convergence etc.
top5.ps:
Metric Topology V, compactness and sequential compactness
a bit of
Hausdorff stuff
top6.ps:
Metric Topology VI, 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:
 [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/pZ
(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 ``padic numbers''
Q_{p} (p prime),
introduced at the end of our topology unit; and more exotic
finite fields such as the 9element field
(Z/3Z)(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_{n} v_{n}
with each v_{i} in V and each
a_{i} in 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.
 Axler does not seem to explicitly use the important notion
of a quotient vector space. If U is a subspace
of a vector space V, we get an equivalence relation
on V by defining two vectors v,v' to be equivalent
(``congruent mod U'') if vv' is in U.
The set of equivalence classes then itself becomes a vector space
(this must be proved!), called the quotient space V/U.
[Note that this is also a notation for V being a vector
space over a field U  we shall strive to make it clear
which meaning we intend when both meanings might make sense.]
 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.
(Axler gives the complexanalytic proof on page 67.)
 Axler unaccountably softpedals the important notion of
duality; we devote much of the seventh problem set
to this, and return to it often later.
 We shall need some ``eigenstuff'' also in an infinitedimensional
setting, so will not assume that any vector space is (nonzero) finite
dimensional unless we really must.
 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_{n}=V, with each
V_{i} of dimension i and containing
V_{i1}. A basis
v_{1}, v_{2}, ..., v_{n}
determines a flag: V_{i} is the span of the first
i basis vectors. Another basis
w_{1}, w_{2}, ..., w_{n}
determines the same flag if and only if each
w_{i} is a linear combination of
v_{1}, v_{2}, ..., v_{i}
(necessarily with nonzero v_{i} coefficient).
The standard flag in F_{n} is the flag
obtained in this way from the standard basis of unit vectors
e_{1}, e_{2}, ..., e_{n}.
The punchline is that, just as a diagonal matrix is one that respects
the standard basis (equivalently, the associated decomposition of
V as a direct some of 1dimensional subspaces),
an uppertriangular matrix is one that respects the standard flag.
Note that the ith diagonal entry of a triangular matrix
gives the action on the onedimensional quotient space
V_{i}/V_{i1}
(each i=1,...,n).
 As explained in class, the generalization to arbitrary fields
F of Axler's treatment (pages 9193) of invariant subspaces
on real vector spaces looks like this:
Suppose u is a nonzero vector in an Fvector space
V, and Q is an irreducible polynomial of degree
d>0 in F[X] such that (Q(T))u=0.
Then the d vectors u,Tu,T^{2}u, ...,
T^{d1}u span a Tinvariant subspace
U of V. Our key claim was that these vectors
are in fact linearly independent, and thus that U is
ddimensional. Indeed, a purported linear relation
would take the form (R(T))u=0 for some polynomial
R, not identically zero, of degree strictly less than
d. But then Q and R are relatively prime,
so there exist polynomials A,B such that AQ+BR=1.
Thus A(T)Q(T)+B(T)R(T) is the identity operator on V.
Evaluating at u yields 0=u, contradiction.
To put this AQ+BR trick in context: let I_{u}
be the set of polynomials P such that P(T)u=0.
Then I_{u} is an ideal in
F[X]: if P,P' are in I_{u}
then so is P+P', as well as AP for any
polynomial A. Now for any polynomial Q, the
set of all polynomials of the form AQ is an ideal, the
socalled ``principal ideal'' generated by Q. Conversely,
every ideal in F[X]
is principal. This is proved as it is for
ideals in Z, using the division algorithm
(Axler, 4.5 on page 66 ff.). This means than if an ideal I
contains an irreducible polynomial Q 
and thus contains the principal ideal generated by Q 
then either I is that principal ideal, or it is all
of F[X]. Going back to our case
I=I_{u}, we find that in the first
case U has dimension d, and in the second
case I_{u} contains 1 so u=0.
[Alternatively, given Q and R,
the set of all polynomials of the form AQ+BR
is again an ideal, etc.]
 A bilinear pairing on a vector space V is said
to be skewsymmetric (a.k.a. ``antisymmetric''
or ``alternating'') if <x,x>=0 for all
x in V. By expanding
<x+y, x+y>
we deduce that <x,y> =  <y,x>
for all x, y in V, whence the terminology.
Conversely the identity
<x,y> =  <y,x>
implies <x,x>=0 (let y=x),
except when the ground field F has characteristic 2,
in which case <x,y> =  <y,x>
amounts to the condition that the pairing be symmetric.
 A norm on a real or complex vector space
V is any function v > v from
V to the real numbers such that:
 The identity cv = c v
holds for all scalars c and all vectors v;
 d(x,y) := xy
defines a distance function on V.
The second condition means that: v is nonnegative
for all v, and zero iff v=0; and
v+v' <= v + v'
for all vectors v, v'. (These correspond
to the nonnegativity and triangle inequality respectively;
what happened to symmetry?) For instance, we have seen in
effect that x := max(x_{1}, ...,
x_{n}) defines a norm on
R^{n} or
C^{n}; on pages 102105,
Axler in effect verifies that x^{2} =
(x,x) gives a norm on an innerproduct space.

Here's an outline of the standard proof of
the Spectral Theorem for selfadjoint oprators on a finitedimensional
real inner product space.
 Here are basic definitions and constructions involving
groups that give a more general context
for some of our results so far and will be especially useful in
our coming discussion of determinants.
 Here are definitions and basic results concerning
cofactors and minors of a matrix.
 And here are some symple results
and pfacts about symplectic spaces and the Pfaffian.
Here are some practice problems
for the final exam, covering both topology and linear
algebra (sometimes both in the same problem). These
are intentionally harder than I expect the final to be.
The following remarks concerning ``little o notation''
are relevant to the concepts of differentiability etc.; cf. the
first ``Remark'' in Rudin, p.213. If f,g are functions
on the same metric space, the notation ``f=o(g) as
x approaches x_{0}'' means:
for every positive epsilon there is a neighborhood of
x_{0} on which f(x) <= epsilon g(x).
Note that for this to make sense, g had better be a nonnegative
real function, and f must take values in a normed vector
space  though equivalent norms yield the same meaning for
``f=o(g)''. Thus: F has derivative F'(x)
at x if and only if
F(x+h)=F(x)+F'(x)h+o(h)
as h approaches 0. (Here F is a vectorvalued
function defined on some neighborhood of x.)
The advantage of this is that we don't have to fiddle
with the special case h=0.
Check that: if f=o(h) and g=o(h) then
f+g=o(h); if f=o(h) and g is
a bounded scalarvalued function then
fg=o(h); if f=o(g) and g=o(h) then
f=o(h); if x is a function of y
continuous at y_{0}, and f=o(g)
as x approaches x(y_{0}),
then f=o(g) as y approaches y_{0}.
The chain rule for vectorvalued functions (Rudin., p.214, Theorem 9.15)
then becomes clear.
[Incidentally, there's also a ``big O'' notation:
``f=O(g)'' means ``there exists a constant C
such that f(x) <= C g(x) for all x''.
Note that this is conceptually simpler since there is no choice
of epsilons.]
p1.ps:
First problem set: Metric topology
p2.ps:
Second problem set: Metrics, topologies, continuity, and sequences
p3.ps:
Third problem set: Sequences cont'd; compactness start'd
p4.ps:
Fourth problem set: Completeness, and compactness: the grand finale
p5.ps:
Fifth problem set: Vector space basics
p6.ps:
Sixth problem set: Bases, dimension, etc.
p7.ps:
Seventh problem set: Linear maps; duality and adjoints;
a bit about field extensions.
p8.ps:
Eighth problem set: Duality, cont'd; eigenstuff;
a bit about bilinear forms and general norms.
p9.ps:
Ninth problem set: Innerproduct spaces, and more about duality.
p10.ps:
Tenth problem set: more about inner products; normal and selfadjoint
operators; a bit about permutations and determinants.