MATH 251-LINEAR ALGEBRA-SPRING 2010- COURSE LOG
Th 1/14 Course policies
Basic
concepts: vectors in R^n, dot product, matrices, action of
matrices on vectors (row and column descriptions),
linear combinations, linear mappings and matrices
Tu 1/19 Basic concepts:
subspaces of R^n (defining equations), geometric
vectors, general vector spaces
Th 1/21 Basic concepts: subspace spanned by a set, linear
independence, basis of a subspace/linear systems in matrix form
consistency and determinacy of systems (definition)
Homework
set 1 (due 1/28)
HW
1 solutions
Tu 1/26 Linear systems- consistency, Range and column space/
Kernel, uniqueness, form of the general solution.Solution set.
Th 1/28 Row space/ orthogonal complement (examples)/
Ker(A) is the orthogonal complement of Ran(A)/ R^n is the direct sum of
E and E^{perp}
rank of
a matrix/ row rank equals column rank (with proof)
Homework
set 2 (due Feb. 4)
HW2
solutions
Tu 2/2 Transpose matrix/ co-kernel and co-range/
example: matrix with given range and kernel/ row equivalence and row
reduction
Basic
dimension theorems
This handout
includes complete proofs of dim Row(A)=dim Col(A) and dim
Row(A)+dim Ker(A)=n. Read it carefully and make sure you
understand the argument.
Th 2/4 Row reduction: examples, applications
Homework
set 3 (due Feb. 11)
HW3
solutions
Tu 2/9 Row reduction: applications, reduced row echelon
form/maximal rank/square matrices and symmetric matrices/ invertible
linear maps; right inverse is also a left inverse.
Th 2/11 Computation of the inverse matrix/ Composition of linear
maps and matrix products: column, dot product and row descriptions.
Homework
set 4 (due TUESDAY Feb. 16)
HW4
solutions
NOTE: The first test will be on
Thursday, Feb. 18. (Included: material up to 2/11 lecture).
Wednesday,
4:30-5:30, Aconda Court 113A: Q&A session (based on the HW sets)
Tu 2/16 Invertible matrices: determinant and inverse
of 2X2 matrices, inverse of a product/ application examples:
interpolating a parabola, evolution by a stochastic matrix, electric
circuits (Ohm's law, Kirchhoff's laws)
Th 2/18 Exam
1
Exam
1-solutions
Tu 2/23 Linear
transformations defined geometrically (final version- revised 2/25)
coordinate vector in a given basis, matrix of a linear
transformation in a given basis/change of basis formulas
Exam 1 returned
Th 2/25 Examples (from handout): projections and
contractions/expansions in R^3/Concept of eigenvalues/change of basis
formula for rectangular matrices
Homework
set 5 (due Mar.4)
HW5
solutions
Tu 3/2 Equivalence
relations for matrices (includes: LU and LPU
decompositions, matrix groups, trace of a square matrix)
(Revised 2010 version,
posted 3/2) Reading guide: Section 1, Example 2 in Section 2,
Section 3 excluding Example 3, Section 4 up to the middle of page 9.
This is all that was seen in class. Section 5 will be discussed later.
Th 3/4 Orthonormal
basis, orthogonal projections and Gram-Schmidt
(On 3/4: up to
p. 3 in this handout)
Homework
set 6 (due March 18)
HW
6 solutions
Tu 3/9, Th 3/11: SPRING BREAK (no classes)
Tu 3/16 Orthogonal group/rotation and reflection
matrices in R^3
Orthogonal
matrices
Th 3/18 Least square solutions, normal system, formula
for projections/linear regression, quadratic regression.
See:
Anton, section 9.3 (9th. edition)
Homework
set 7 (due March 23)
HW7
solutions
Tu 3/23 determinants
Notes
on determinants
(preliminary version; fall 2009)
Th 3/25 Exam
2 (inc. material up to 3/18- HW sets 5,6,7)
exam
2-solutions
Tu 3/30 characteristic polynomial/ eigenvalues and
eigenspaces of 2x2 matrices- complex eigenvalues (start)
[powers
of matrices and similarity, square matrices as dynamical systems]
Complex
eigenvalues of real matrices
REVIEW:
complex numbers
Th 4/1 standard
forms of square matrices (n=2,3) (includes: limits of powers)
HW set 8: the 10 problems at the end
of this handout (due: 4/8)
HW8
solutions
Tu 4/6 similarity classification/ principal axes
theorem (start)
Th 4/8 principal axes theorem/ quadratic
forms (start)
HW set 9: problems 0 to 8 in
this handout (due 4/15)
Principal
Axes Theorem (includes quadratic forms, singular value
decomposition)
Remark: the proof of Liouville's formula may be found in the
handout "Notes on determinants"
Tu 4/13 quadratic forms/ singular value decomposition
(start)
Th 4/15 singular value decomposition (end)/difference
equations (start)
HW set 10: problems 9, 10 ,11 in the
handout "Principal Axes Theorem" (due 4/20)
HW9
and HW10-solutions
Tu 4/20 Fibonacci example/ limits of powers (see handout
"standard forms")
Solution
of linear recursions
(W 4/21: optional review session)
Th 4/22 Exam 3 (lectures
from 3/30 to 4/15)
Exam3
solutions
Tu 4/27, Th 4/29 stochastic matrices/ Perron-Frobenius theorem
stochastic
matrices
FINAL EXAM: The final will consist of 8 problems; 6 will be based on (not identical)
questions on exams 1,2,3 (two each), and two
will be based on the material introduce since exam 3 (linear recursions
and stochastic matrices; see online handouts for problems).
I'll be available by appointment (email) to answer questions during
finals week.
Final dates:
SECTION 3 (11:10)- Wednesday, May 5, 10:15-12:15, HBB 102
SECTION 4 (9:40)-Friday May 7, 8-10, HBB 130
(the usual classroom, in both cases)
Final
Exam A
Final
Exam B