Math 314 log, spring 2016


Wed Jan 13
sets and functions, injections, surjections, bijections, cardinality, 
partial ordering, chains (=lineared ordered sets), maximal elements, 
Zorn's lemma.

 
Friday Jan 15
equivalence relations, equivalence classes, mathematical induction, 
definition of fields.


Friday Jan 22
definition of vector spaces over a given field, subspaces, 
solution of a system of linear equations as an example of subspaces, 
matrices over a field, elementary row operations, row reduced matrices,
row-reduced echelon matrices, row equivalent matrices, matrix
multiplication, invertible matrices, upper triangular and lower triangular
matrices, symmetric matrices, linear combinations, vector subspaces.


Monday, Jan 25
linear span, linear independence, bases.

Friday, Jan 29
proof that any two finite bases of a vector space have the same cardinality
(i.e. dimension of a finite dimensional vector space is well-defined),
linear coordinates, change of bases

Monday, Feb 1
change of bases continued, linear transformations


Friday, Feb 7
matrix representation of a linear transformation, change of basis,
the space of all linear transformations between two vector spaces,
the dual of a vector space


Monday, Feb 8
dual vector space, dimension of the dual of a finite dimensional
vector space, the double dual


Friday, Feb 12
transpose of a linear transformation, matrix representation
of a transpose


Monday, Feb 15
more one the double dual, duality relating the kernel and image
of a linear transformation and the kernal and image of the transpose


Friday, Feb 19
summary of transpose, dual and double dual.
Recall the definition of polynomial in one variable,
long devision.


Monday, Feb 22
polynomial ring in one variable over a field, the PID property,
how to think in terms of ideals (instead of divisor/multiple)


Friday, Feb 26
application of the PID property to polynomials in a give linear operator,
minimal polynomial


Monday, Feb 29
how to use the minimal polynomial to produce projectors when the minimal
polynomial has more than one irreducible factor, permutations,
cyclic permutations, disjoint cycle decomposition,
definition and properties of the sign of a permutation


Frdiay, March 4
properties of the determinant function: as a homogenous polynomial,
multi-linear functions, alternating functions, uniqueness
and existence


Monday, March 14
properties of determinants, space of all alternating multi-linear forms


Friday, March 18
in-class exam


Monday, March 21
definition of modules, main example: modules over F[x] where F is a field

Friday, March 25
eigenvalues, eigenvectors, 
How to use the prime factorization of the minimal polynomial of an operator
to decompose a vector space into direct sum of subsapces stable under the
operator

Monday, March 28
characteristic polynomials, Cayley-Hamilton


Friday, April 1
review of the proof of Cayley-Hamilton theorem
Invariant subspaces, simultaneously diagonalizable family of linear operators,
simulataneously triangularizable family of linear operators

Monday, April 4
direct sum decomposition, comuting mutually orthogonal idempotents, invariant
direct sums, primary decomposition

Friday, April 8
Jordan canonical forms, and its intrinsic formulation, cyclic subspaces
(for a given linear operator)

Monday, April 11
rational canonical forms, statement and its intrinsic formulation
nilpotent operators, structure of 
triangularizable operators

Friday, April 16
two equivalent formulations of the theorem on rational canonical forms,
inner products, Cauchy-Schwartz inequality, Gram-Schmidt process,
orthogonal basis


Monday, April 18
How to identify the dual of a finite dimensional vector space with the
vector space itself , ajoint of a linear transformation,
definition of normal operators, unitary operators,
hermitian operator, skew hermitian operators,
statement of the spectral theorem for normal operators.


Friday, April 22
Proof of the spectrual theorem of normal operators,
applications


Monday, April 25
Review