Class Diary for M251, Spring 2010, Jochen Denzler
Wed Jan 13:
Formalities; Systems of linear equations; coeff and augmented matrix.
Fri Jan 15:
Easy matrix vocabulary (row, column, etc);
elementary row operations, row echelon form and Gauss elimination
Mon Jan 18:
MLK DAY
Wed Jan 20:
backward substitution. --
Gauss Jordan elimination and reduced row echelon form. Adding and multiplying
matrices. Hwk: Ch 1, numbers 1-9 due Mon (refer to handed-out hwk list, not
to book for hwk)
Fri Jan 22:
Matrix multiplication and all the rules of matrix arithmetic; in particular
non-commutativity: AB may differ from BA. Identity matrix.
Mon Jan 25:
Definition of inverse matrix, and how it relates to solving SLE's if we have
it. Calculation of inv matrix outlined.
Wed Jan 27:
Detailed calculation of inverse matrix; uniqueness; Hwk Ch 1. 11-19 due Mon
Fri Jan 29:
Elementary matrices and their role in claculating inverse matrices
Mon Feb 01:
Triangular, Diagonal, and symmetric matrices; and their products. General
remarks on symbolic matrix calculations. Hwk: 20-25 due Fri, 26-31 due
Mon
Wed Feb 03:
(1) product of symmetric matrces is symmetric if and only if they commute. ---
(2) LU decomposition.
Fri Feb 05:
Vectors in space and in the plane; geometric vs algebraic interpretation;
correspondence between the two, given a cartesian coordinate system.
addition of vectors, and multiplication of vectors by numbers. Norm.
Hwk: Read notes on points, vectors, and origins
handed out
Mon Feb 08:
Dot product of vectors; geometric and algebraic. Connection via law of
cosines in a triangle. Work as a dot product. You can already start Hwk 1,2
for Ch 3. Turn-in time later, when we have a few more.
Wed Feb 10:
Examples how to calculate angles with the dot product. Cross product defined
geometrically and algebraically. Hwk Ch 3, 1-10 due Mon (may need a few
hints on #9,10 yet)
Fri Feb 12:
cross product and scalar triple product; geometric meanings, including oriented
volume and area; questions and answers on hwk
Mon Feb 15:
Introduction to determinants, in view of the special cases in for 2x2 and
3x3 matrices. n! many products of n entries each, some permutations get + and
some -
Wed Feb 17:
Even and odd permutations: by inversions and by swaps (see glossary for swaps);
quick review for exam. %%%% Q&A session for exam will be tomorrow 5pm
in AC room 113 %%%% determinant of a triangular matrix calc'd easily.
Fri Feb 19:
EXAM 1
Mon Feb 22:
How elementary row operations affect the value of determinants. Exam returned
Wed Feb 24:
Calculation of determinants via row operations. Also det(A^T)=det A, so you may
also use column operations for determinants, if convenient. Hwk: Ch.2, 1-9
due Mon
Fri Feb 26:
Minors, cofactors and adjoint matrix defined. A.(adj A) = (det A) I
observed in one example.
Mon Mar 01:
Row and column expansion of determinants.
Wed Mar 03:
Explained why A.(adj A) = (det A) I. Cramer's rule.
Fri Mar 05:
Characteristic polynomial of a matrix; trace as coefficient of 2nd highest
power in char. polynomial
Mon Mar 08:
SPRING BREAK
Wed Mar 10:
SPRING BREAK
Fri Mar 12:
SPRING BREAK
Mon Mar 15:
R^n, its arithmetic including dot product and norm; Cauchy Schwarz inequality
and angle measurement in R^n
Wed Mar 17:
Triangle inequality proved.
Linear mappings from R^n to R^m and their graphic representation in the case of
R^2 to R^2. Ch 4, Hwk 1-6 due Mon
Fri Mar 19:
Matrices of reflections and projections. The role of eigenvectors and
eigenvalues Hwk Ch 4, 7-8 due Wed
Mon Mar 22:
The role of the determinant as area (or volume) mapping ratio.
Rotation matrices in the plane. Composition of mappings corresponds to
multiplication of matrices (watch for the correct order)
Wed Mar 24:
Orthogonal matrices, ratations in space; eigenvector Hwk Ch 4, 9-11 due Mon
Fri Mar 26:
Real vector speces: motivation, definition, examples; subspaces
Mon Mar 29:
Vector space language: linear combination; span; linearly dependent;
Wed Mar 31:
Calculationally checking linear independence and spanning in R^n. Basis.
Fri Apr 02:
GOOD FRIDAY
Mon Apr 05:
EXAM 2
Wed Apr 07:
Theory concerning lin indep, basis, spanning. Dimension defined.
An example of linear independence in a space of functions Hwk: Ch.5,
pblms 0-7 due Mon
Fri Apr 09:
More on linear dependence and independence of functions; vsp of polynomials,
Hwk 8-15 due Fri (will need to comment on #12 on Monday yet)
Mon Apr 12:
solutions of homogeneous linear equations form vector spaces; null space of a
matrix.
Wed Apr 14:
Row and column space of a matrix and how to find bases for them; rank. Hwk
16-22 due Mon
Fri Apr 16:
Questions re pending homework; Rank and Nullity; their sum = number of
columns.
Inner product defined.
Mon Apr 19:
The basic examples of inner products in vector spaces. Hwk: Ch.~6, 1-7 due
Fri (re 6 and 7 need a bit info on Wed yet)
Wed Apr 21:
Cauchy Schwarz inequality and angle. Review of eigenvalues, characteristic
polynomial.
Def of algebraic and geometric multiplicity of an eigenvalue.
Fri Apr 23:
Simple example where geom and algebraic multiplicity differ. Sum of eigenvalues
is trace. Product of ev's is determinant. --- Review for exam and TCE.
Mon Apr 26:
EXAM 3
Wed Apr 28:
All about eigenvalues and eigenvectors reviewed; and the diagonalization problem
Fri Apr 30:
Diagonalization, in particular, of symmetric matrices; and application in
physics of rotating objects. Hwk for Ch 7 will not be graded, but is good
for training and solutions are posted.
Mon May 03: STUDY DAY
Fri May 07: FINAL EXAM 10:15-12:15
(scheduled by university policy)
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