Class Diary for M251, Spring 2019, Jochen Denzler
Wed Jan 09:
Intro: Systems of linear equations; matrices.
Fri Jan 11:
elementary row operations, Gauss elimination and row echelon form; examples;
Gauss-Jordan and reduced row echelon form (briefly)
Mon Jan 14:
An example for Gauss-Jordan and r.r.e.f. Operations with matrices defined.
Homework due Friday: Numbers 1-9 from my Hwk list.
Wed Jan 16:
algebraic rules for matrix operations; identity matrix
Fri Jan 18:
SLE's and matrix multiplication. Inverse matrix defined and how to calculate
it in practice.
Mon Jan 21:
MLK Day
Wed Jan 23:
Properties of inverse matrices, eg, (AB)^(-1)=B^(-1)A^(-1). SLE via inverse
matrices. -- Elementary row ops via inverse matrices (just started).
Hwk: numbers 10-17 from first hwk list due
Monday.
Fri Jan 25:
Elementary matrices; they're invertible. Proof AB=I implies BA=I. Preview of LU
decomposition.
Mon Jan 28:
LU decomposition: How to do it and why it's useful. Properties of SLE's with
invertible and with singular coefficient matrices.
Hwk: numbers 20-26, 32 from first hwk list due
Friday.
Wed Jan 30:
triangular, diagonal, and symmetric matrices. Product of symmetric matrices is
symmetric if and only if they commute. Trace. Trace(AB)=Trace(BA).
Fri Feb 01:
Vectors in R^2, R^3, and R^n. Geometric view as well as algebraic view (as
single-column matrices). Linear operations on vectors. Hwk: remaining
problems 18,19,27-31 from 1st sheet due
Wednesday.
Mon Feb 04:
The dot product, geometric and algebraic; its properties; Cauchy Schwarz
inequality
Wed Feb 06:
(some hwk feedback); dot product and law of cosines; cross product defined
algebraically, and its key properties. All homework
from next sheet (Chapter 3) due next Wednesday.
Fri Feb 08:
Orientation of cross product (right hand rule), and scalar triple product;
oriented volume. 3x3 determinant (about to be defined) is a scalar triple product.
Mon Feb 11:
nxn determinants; promise to represent an oriented `hyper'volume of boxes in
R^n. Analog from 2x2 and 3x3. Formula in these cases, and general
(combinatorial) formula
started.
Wed Feb 13:
Determinants by combinatorial formula: how to do the + and - right by either
inversions or swaps; key properties how det's are impacted by elementary row
operations; practical example. NOTE: I will post a copy from an older
version of your textbook with the chapter covering the combinatorial formula
on canvas shortly, in case you missed the hardcopy: Look for
`OldDetChapter.pdf' under `Files'
Fri Feb 15:
geometric interpretation of the determinant rules.
Mon Feb 18:
Row and column expansions (and how to use them wisely).
Wed Feb 20:
EXAM 1
Fri Feb 22: Rules about determinants; in particular det AB = det A det B
(proved). No rule for det(A+B).
Hwk: Problems 1-11 of set for Ch. 2 due next
Friday, the rest by Monday Mar 04
Mon Feb 25:
exam discussion; Cramer's rule stated, and examples and use.
Wed Feb 27:
cofactor matrix and adjoint of a matrix; inverse matrix by determinants; why
this works; and why Cramer's rule works as a consequence.
Fri Mar 01:
Geometry of lines, planes, solutions of SLEs in R^2, R^3, R^n. Superposition
principle. (This refers to Ch 3.4 of book, which I had previously skipped)
Mon Mar 04:
Intro to General vector spaces: Motivation, axioms and examples.
Wed Mar 06:
More examples; Subspaces;
Fri Mar 08:
Span; determining whether a vecor is in the span of a set
Mon Mar 11:
Linear independence: Examples; at most n lin indep vectors in R^n; basis of a
vector space.
Wed Mar 13:
Examples of bases; coordinate vector. Hwk 1-10
from this list
due Wed after spring break.
Fri Mar 15:
Example how to find a basis for a plane. Kernel / Null space. Theory on bases.
Mon Mar 18:
SPRING BREAK
Wed Mar 20:
SPRING BREAK
Fri Mar 22:
SPRING BREAK
Mon Mar 25:
Review and examples for bases and coordinates of a vector with respect to a
basis; null space, column space, and row space of a matrix: definition and some
examples.
Wed Mar 27:
Calculation of bases for null space, row space, and column space.
Fri Mar 29:
Rank and nullity of a matrix. Rank(A)+nullity(A)=number of columns of A.
Intro to eigenvalues and eigenvectors (see Ch 5.1- will return to rest of Ch 4
after exam). Hwk: Pblms 11-20 from this list due next
Friday.
Mon Apr 01:
EXAM 2
Wed Apr 03:
discussion of exam; the function f(x)=Ax defined by a matrix A in simple
examples.
Fri Apr 05:
Dilations, Reflections, 2D rotations
Mon Apr 08:
(orthogonal) Projections; geometric meaning of determinant, eigenvalues and
eigenvectors. Hwk: Problems 1-8 from this list due
Friday. For Pblm 1-6, use one copy each of this
sheet.
Wed Apr 10:
eigenvalues and vectors and their geometric interpretation reviewed;
composition of linear mappings and matrix product. Rotations in R^3 and
orthogonal matrices. Hwk: Problems 9-11 from this
list due Monday.
Fri Apr 12:
rotation axis = eigenvector for eigenvalue 1 of rotation matrix.
Characteristic polynomial.
Mon Apr 15:
Wed Apr 17:
Fri Apr 19:
GOOD FRIDAY
Mon Apr 22:
EXAM 3
Wed Apr 24:
Fri Apr 26:
Mon Apr 29: STUDY DAY
Wed May 01: FINAL EXAM 10:15-12:15
(scheduled by university policy)
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