Class Diary for M251, Fall 2013, Jochen Denzler
Wed Aug 21:
Intro. Linear equations. Systematic elimination procedure. Coefficient and
Augmented matrix
Fri Aug 23:
Matrices and names for parts thereof. Elementary row operations and row echelon
form, and how they relate to elimination from equations. Hwk: 1-9 due Wed.
You need to read up on Gauss and Gauss-Jordan in book (or wait for Mon) do do
numbers 4,5,
Mon Aug 26:
Gauss-Jordan and reduced row echelon form. Sneak preview about Matrix algebra.
Def of matrix addition and multiplication. Hwk: read up on transpose of
matrix
Wed Aug 28:
Hints re hwk pblm 6; extension as needed. Writing SLE's as matrix equations.
Algebra of matrix operations, non-commutativity. Hwk: 11-19 due Wed (ignore
2nd paragraph `Also write...' in pblm 16)
Fri Aug 30:
Proof of associative law for matrix multiplication; identity matrix. Def. of
inverse matrix and preview of facts about it.
Mon Sep 02:
LABOR DAY
Wed Sep 04:
Inverse matrix: uniqueness; how to calculate in practice.
Fri Sep 06:
Example of inv matrix calculation; Properties of inverse matrix; elementary
matrices begun. Hwk 20-26 due Wed
Mon Sep 09:
Elementary matrices used; If AB=I, then BA=I automatically (for square
matrices). Quick preview LU decomposition
Wed Sep 11:
LU decomposition in detail
Fri Sep 13:
Theoretical issues about the inverse. (Ax=b uniquely solvable if and only if A
is invertible). Hwk 32 due Wed
Mon Sep 16:
Thm 1.6.4 from book. Symmetric matrices. Geometric vectors in plane and space,
and how to add them. Hwk 27-32 due Fri (32 extended from Wed for logistics
reasons)
Wed Sep 18:
Geometric vectors and their identification with 2x1 or 3x1 matrices. Dot
product introduced.
Fri Sep 20:
Dot product algebraically vs geometrically. Example for calculating angles.
Cross product quickly defined algebraically; occurrences in physics mentioned.
Hwk: Consider all problems from the hwk sheet for Ch 3 as assigned.
They will not be collected for grading. I'll post solutions soon. As of now
you're able to do all problems but #9. The ones with the cross product will
come easier after Monday. -- Also read the notes on `points vs vectors' as
handed out today, and posted on the website
Mon Sep 23:
Geometric characterization of cross products. Right hand rule. Cross product as
area of parallelogram. Scalar triple product started as volume of
parallelepiped.
Wed Sep 25:
Geometric role of scalar triple product explained in detail; review for exam;
1x1, 2x2 and 3x3 determinants arising as formulas for oriented length, area and
volume respectively.
Fri Sep 27:
EXAM 1 Hwk: make sure to review the 1x1, 2x2, 3x3 determinant notes from the
end of Wed class by Monday
Mon Sep 30:
Quick review of 2x2 and 3x3 determinants and general pattern of nxn
determinants. Abstract definition of nxn determinant as sum/difference of
products; sign of a permutation by either counting inversions, or counting
swaps.
Wed Oct 02:
Basic rules about determinants, specifically under row operations.
Fri Oct 04:
Examples for determinants by row and column operation. Hwk 1-9 due Wed.
Exam back.
Mon Oct 07:
Expansion of rows and columns and how to combine them conveniently with row and
column operations for effective calculation; A invertible exactly if det A not
0.
Wed Oct 09: Q&A.
det (AB) = (det A)(det B) explained. Hwk Ch 1 #33-34 due Mon general
explanations to this theoretical style of questions.
Fri Oct 11:
adjoint of a matrix. Ainverse = adj A / det A
Mon Oct 14:
Cramer's rule, and when it is or is not efficient to use.
Wed Oct 16:
Hwk: remaining problems for Ch.~2 due Wed Introduction to real vector
spaces;
R^n; vector space of mxn matrices; vector space of real functions.
Fri Oct 18:
FALL BREAK
Mon Oct 21:
Vector space of functions, explanations. --- Matrix operations started.
Hwk 1-6 on matrix operations due Monday; more to be assigned Wed.
Wed Oct 23:
composition of matrix operations corresponds to multiplication of matrices.
Reflections started.
Fri Oct 25:
Reflections, Projections and 2dim rotations. Examples where Ax is a multiple
of x. Hwk: remaining problems on matrix operations due Fri
Mon Oct 28:
Eigenvectors and eigenvalues; rotations in space by orthogonal matrices begun
Wed Oct 30:
Orthogonal matrices
Fri Nov 01:
Some hwk hints; vector spaces resumed (beginning of Ch. 4 in book); subspaces;
examples. Linear combinations.
Mon Nov 04:
Linearly independent; spanning; basis; dimension
Wed Nov 06:
Examples of Linear independence in various vector spaces. Hwk 0-7 on
Abstract Vector Spaces Ch 4.1-8 (formerly Ch.~5) due Fri Nov 15
Fri Nov 08:
EXAM 2
Mon Nov 11:
Basis and coordinate vector with respect to basis. Exam back.
Wed Nov 13:
(subst) Discussion of Hwk questions
Fri Nov 15:
(subst)
Mon Nov 18:
Eigenvalues; characteristic polynomial; trace equals sum of eigenvalues,
determinant equal product of eigenvalues.
Wed Nov 20:
Calculation of eigenvectors; algebraic vs geometric multiplicity.
Without proof: geom mult <= algebraic multiplicity.
Without proof: symmetric matrices have only real eigenvalues
Fri Nov 22:
Mon Nov 25:
EXAM 3
Wed Nov 27:
Fri Nov 29:
THANKSGIVING BREAK
Mon Dec 02:
Wed Dec 04: STUDY DAY
Tue Dec 10: FINAL EXAM 12:30-02:30
(scheduled by university policy as a function of class meeting time)
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