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