Class Diary for M251, Spring 2008, Jochen Denzler
Wed Jan 09: Systems of linear equations, and matrices. Hwk for Fri:
read Secs 1.1 and 1.2 in book
Fri Jan 11:
Example; Gauss and Gauss-Jordan elimination; row echelon and reduced row
echelon form. Matrix multiplication def'd. Hwk for Mon: read Sec 1.3 of
the book, and fill out the scheduling form so I can schedule convenient
office hours for you. --- Hwk for Wed: #1-9 (ask Mon if problems arise)
Mon Jan 14: Rules of matrix arithmetic. In particular proved (AB)C=A(BC).
Hwk for Fri: #11-18
Wed Jan 16: Inverse matrix of a square matrix. (there can be at most
one) Hwk for Fri: read ch 1.4 and 1.5.
Hwk for next Wed: #19-23
Fri Jan 18: Remarks re hwk.
How to calculate an inverse matrix. Inverse of products.
Hwk: read 1.5
Mon Jan 21: --- MLK DAY ---
Wed Jan 23: Elementary matrices. Hwk 20 due Fri, if not turned in
today. Also read Sec 1.6 for Friday. -- Hwk 24-29 will be due next Wed;
you can do 24 and 25 already now, but the others need new material.
Fri Jan 25:
Properties of linear systems with invertible matrices. Triangular, diagonal,
symmetric matrices.
Mon Jan 28:
LU decomposition.
Hwk for Fri: 30+31 (graded by myself); and *separately* 32 (goes to grader).
Also don't forget Wed due 24-29
Wed Jan 30:
Geometric vectors (arrows) in the plane and in space, and their arithmetic.
Representation as 2x1 or 3x1 matrices (columns). Norm. Hwk: read
sec 3.1-3.3 of book, and the notes
on points vs vectors handed out
Fri Feb 01: Comments on hwk. Dot product; geometric and algebraic
definition, and why they amount to the same. An example for calculating angles
with the dot product. Hwk: Read sec 3.3; skim sec 3.4 (not worrying
about determinant references therein). -- For Wed Hwk 1-4 of the `Chapter 3'
sheet
Mon Feb 04:
Comments re Hwk 30,31 from ch.1 ---
Work as a dot product; projections; algebr def of cross product; its geometric
properties begun.
Hwk: Hwk 5-10 due Fri. #9 needs Wed class yet
Wed Feb 06:
geom properties of cross product; oriented area and oriented volume; scalar
triple product.
Fri Feb 08:
review ori. area and volume; these quantities are special cases of determinants.
Overview on determinants; permutations, even and odd
Mon Feb 11:
Review for exam; Q&A. Determinants defined. Easy cases: zero row; diagonal or
triangular. without proof det A = det (A^T)
Wed Feb 13: EXAM 1
Fri Feb 15:
Determinants by row reduction. Hwk: 1-5 due Wed -- NOTE:
Since the last edition, the book has swapped some sections on determinants.
What I have been starting with is now basically in Sec 2.4 of the 9th edition,
and now we go through 2.2 and 2.3, and we will finally do 2.1
Mon Feb 18:
Row and column expansion for determinants. -- Exam back
Wed Feb 20:
Minors, cofactors and the adjoint matrix; (A) (adj A) = (det A) I
Hwk: 6-11 due Mon
Fri Feb 22:
Cramer's rule: when to use it and when not. det(AB)=(det A)(det B),
and review of all the other det formulas.
Hwk: 12-15 due Wed
Mon Feb 25:
Q&A; discussion of some hwk. R^n and dot product. Cauchy Schwarz inequality.
Quick preview on linear mappings.
Wed Feb 27:
Theoretical remarks: Conclusion direction `if...then'. Linear mappings and
matrices.
Hwk Ch4, 1-6 due Mon
Fri Feb 29:
Comments on determinant homework. --- Matrices representing a reflection.
Hwk: Ch4, 7+10
Mon Mar 03: Eigenvalues and eigenvectors how to calculate them,
and their geometric interpretation. Hwk: Ch4, #8 also due Wed (on top of
7,10 assigned last time)
Wed Mar 05: 2D rotation; the determinant of a mapping matrix.
Fri Mar 07:
composition of two mappings corresponds to matrix product; 3D rotations
and orthogonal matrices. Hwk 9,11 due Wed
Mon Mar 10:
Q&A re rotations and eigenvectors. --- Abstract real vector spaces: definition
and examples. Hwk (informally; not for turn-in: think whether the set of
symmetric nxn matrices forms a vector space; how about the set of orthogonal
nxn matrices?
Wed Mar 12:
Fri Mar 14: SPRING BREAK
Mon Mar 17: SPRING BREAK
Wed Mar 19: SPRING BREAK
Fri Mar 21: GOOD FRIDAY
Mon Mar 24:
Review for exam. Definition of span and linear independence.
Wed Mar 26: EXAM 2
Fri Mar 28:
Examples how to check linear independence. Basis of a vector space defined.
Hwk: Ch. 5, numbers 1,3,6,7. Due Fri (more to come). Read Sec 5.1-5.3 and
into 5.4 of book
Mon Mar 31:
Examples for bases. Exam 2 back
Wed Apr 02:
Coordinate vector. Theorems about bases. Hwk: all of 1-9 due Fri (if
possible); will also be accepted Monday. Read through 5.4 of book for good.
Fri Apr 04:
Row and column space of a matrix. How to find bases for them (and why it works).
Mon Apr 07:
Basis of the null space of a matrix; rank and nullity
Wed Apr 09:
Hints about new hwk. Inner products: definition and examples Hwk: Ch5, 10-20 due Mon
Fri Apr 11:
More examples; in particular those in R^3 where one may need to complete
squares to check the last property.
Mon Apr 14:
Gram Schmidt orthogonalization. Hwk: Ch6, 1-7. To be discussed on Wed
if questions have arisen. Serves as part of the exam prep, but will not
be collected
Wed Apr 16:
Review for exam
Fri Apr 18:
EXAM 3
Mon Apr 21:
Review eigenvalues; char. polynomial; trace = sum of eigenvalues,
det = product of eigenvalues. --- Evaluation.
Hwk: Ch6, 8+9, due Wed. All Hwk Ch 7 is for self-study and will not
be picked up for grading
Wed Apr 23:
Geometric vs algebraic multiplicity of eigenvalues. (geometric mult' is less or equal
alg multiplicity). Problem posed: to find a basis of eigenvectors --- Exam back
Fri Apr 25:
(Writing ahead of time). Diagonalization and similarity. Diagonalization of symmetric
matrices.
Mon Apr 28: STUDY DAY
Mon May 05: FINAL EXAM 12:30-2:30
(scheduled by university policy)
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