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. ==================================== 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. #################################################### 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