COURSE OVERVIEW

Since many lecture topices have stayed relatively close to the book, I 
am refraining from posting such notes on the web, and attempting to post
commentary on tidbits and topics encountered in lectures which are not
readily found in the textbook.

Following E. Artin, P. Lax, and others, I have taken the definition of the
(real) determinant A as n! times the signed volume of a simplex whose n+1
vertices are the n-dimensional vectors located at 0 and at the n columns
of the matrix A.

The first two sections of Chapter 3 cover inner products, lengths, and
orthogonality.  By viewing a determinant as a volume (i.e., 1/n times
base times altitude), we are able to begin our study of determinants by
applying these notions immediately.

Hopefully, this motivates a few abstract properties such as multi-linearity
which can be postulated, and from which (as in the book) the other properties
of determinants can be derived.  The slightly later section 4.4.3 of the book
presents the determinant as the volume of an appropriate n-dimensional 
parallelpiped.  This avoids the factor of n!, but requires a polytope of
2^n vertices instead of one with only n+1.  Otherwise, the views are
almost equivalent.  The other (minor) difference is that in section 4.4.3,
we transpose the matrix and work with row vectors instead of column
vectors.

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SYLLABUS OF SOME INTRODUCTORY LECTURES


Aug 24, 26:  Introduction, review of complex numbers, exp 2 pi i = 1
	Formal definition of field.  Fields: Q, R, C, and Z2

Aug 31: Definitions and some properties of matrices.
	Examples of strictly lower triangular matrices as nilpotent.
	Winograd's (recursive!) formula for multiplying a pair of
	2x2 matrices (whose elements may themselves be square matrices)
	in 7 multiplications and 15 additions (instead of 8 multiplications
	and 4 additions):


	a  b    A  C    =     aA + bB                   w+(c+d)(C-A)+(a+b-c-d)D
	c  d    B  D          w+(a-c)(D-C)-d(A-B-C+D)   w+(a-c)(D-C)+(c+d)(C-A)

	where w = aA - (a-c-d)(A-C+D)
	
	
Sep 2: Elementary matrices, permutation matrices, factorization
	into lower triangular and upper triangular...
	
	(I+A)(I+B) = I + A + B + AB

	Rudiments of numerical accuracy: 
		1) traditional formula for roots of quadratic equation
		2) matrix with small pivot
	

The lectures on Sep 9, 14, and 16 covered the following
topics, all of which also appear in the textbook:



Axioms for a vector space.

v. Rudimentary theorems about vector spaces, w proofs:
  e.g.,(scalar 0) v = (vector 0)

Examples of vector spaces

Fundamental subspaces of a matrix:
	Row space
	null space
	column space
	left nullspace

Adjacency matrix of a directed graph, with columns
 coordinatized by nodes, and rows by branches.

 Lemma: A set of rows in indepent iff the corresponding
 branches are loop-free.

 Spanning trees



Elementary geometric transformations of R^2 viewed as
2x2 matrices:

	Expansion/Contraction
	Rotation
	Reflection
	Projection

There was also further review of the LU factorization,
with more emphasis on a careful matrix-level formulation
of the procedure.  We saw that getting from one line,

	M = A B

to the next,

	M = C D

often involves the insertion of a pair of matrices in
the middle:

	M = A B = A E^-1 E B
		= (A E^-1) (E B)

where E might be an elementary matrix (equal to the 
identity plus a single off-diagonal element, whose
inverse is the same except for a sign-change of the
off-diagonal element),  or a transposition, which is
a permutation whose inverse is itself.  If T is such
a transposition, then after writing

	M = A B = A T T B

we usually continue to

	TM = TAT TB  = (TAT)(TB) = C D

So as the factorization procedure continues, we may
discover a need for further transpositions (in order to
bring big or at least nonzero elements into successive
pivot positions), and these the product of these
successive transpositions becomes the permutation matrix
P which premultiplies M.  If we are lucky or clairvoyant,
we might avoid any such permutations, but just as we
may be many computations before discovering that the
matrix is singular, in some cases we might have to do
many computations before discovering that some
permutation is required in order to do the LU
factorization.

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Other lecture notes posted on this website appear in
files whose titles are abbreviated as follows:

L.U       = Factorization of matrix into Lower and
             Upper Triangular Matrices

ECC       = (binary) Erasure Correcting Codes

Euclid.Rn = Euclidean Geometry in n Dimensions

FFT       = Fast Fourier Transform