Meeting Time  Mon., Wed., Fri.,
2PM  2:50PM 
Location  CENTER 113 
Instructor 
Dragos Oprea

Course Assistants 
Christopher Briggs
Daniel Minsky

Textbook  Gilbert Strang, Applied Linear Algebra, 4th
edition.
Required. Available at the bookstore and on reserve in the library.

Grade Breakdown  The grade is computed as the following
weighed average:  Homework 20%, Midterm I 20%, Midterm II 20%,
Final Exam 40%.

Course Content 
Math 102 can be thought of as a sequel to Math 20F. The topics
covered will include row reduction, LU decomposition, the four spaces
associated to a
matrix, projections, GramSchmidt, QR decompositions, least
squares, eigenvalue problems,
difference equations, abstract vector spaces, linear
transformations, abstract inner products,
Hermitian matrices, singular value decomposition, pseudoinverses.

Prerequisites  Math 20F or Math 31AH. 
Readings  Reading the sections of the textbook
corresponding to
the assigned homework exercises is considered part of the homework
assignment. You are responsible for material in the assigned reading
whether or not it is discussed in the lecture. It will be expected that
you read the assigned material in advance of each lecture. 
Homework 
Homework problems will be assigned on the
course
homework
page. There will be 7 problem sets, due certain Wednesdays at 4:30pm
in
the TA's mailbox. You may work together with your classmates
on your
homework
and/or ask the TA's (or myself) for
help on assigned homework problems. However, the work you turn in must be
your own. No late homework assignments will be accepted.


Midterm Exams  There will be two midterm exams given
in
class. The dates are January 28 and February 27. There will be no makeup
exams. 
Final Exam  The final examination will be held on
Monday, March 18, 36 PM. There is no make up final
examination. It is your responsability
to ensure that you do not have a schedule conflict during the final
examination; you should not enroll in this class if you cannot
sit for the final examination at its scheduled time. 
Announcements and
Dates 
 Monday, January 7: First lecture
 Monday, January 21: MLK Day, no class.
 Wednesday, January 28: Midterm I
 Monday, February 18: Presidents' Day, no
class.
 Wednesday, February 27: Midterm II
 Friday, March 15: Last Lecture
 Monday, March 18: FINAL EXAM, 36PM.

Exams 

Lecture
Summaries 
 Lecture 1: Introduction. Elimination. Row operations. Row
reduction. Row echelon form. Row reduced echelon form.
 Lecture 2: Effect of row operations on a matrix. Elementary
matrices. LU
decomposition. Existence and uniqueness. LDU and PLU decomposition.
 Lecture 3: LDL decomposition of symmetric
matrices. Using LU
decompositions to solve systems. Solving
systems via inverses. Computing inverses via row reduction.
Homogeneous vs. inhomogeneous systems.
 Lecture 4: Abstract vector spaces and subspaces. Examples:
spaces of
functions, spaces of polynomials, spaces of matrices, spaces of
sequences, solutions to
differential equations, and others.
 Lecture 5: Linear independence, span, bases, dimension for
abstract vector spaces. Properties and examples.
 Lecture 6: The four subspaces associated to a matrix and
their dimensions. Column
spaces, null space, row space, left null space. Rank and nullity. Finding
bases for the four
subspaces.
 Lecture 7: Existence of left and right inverses. How to
compute left and right inverses. Examples.
 Lecture 8: Abstract linear transformations and examples.
Rotations, projections, reflections, differentiation, integration. Matrix
of a linear transformation in a given basis.
 Lecture 9: Matrix of an abstract linear transformation.
Reading the four subspaces of an abstract linear transformation from
matrix
representation. Midterm review.
 Lecture 10: Dot product and orthogonality. Properties of dot
product. Orthogonal complements and properties. Orthogonality and the four
subspaces of a
matrix.
 Lecture 11: Matrix of projections and the left inverse. Least
squares
solutions. Weighted least squares. Best fit line. Examples. How does your
GPS work?
 Lecture 12: Orthonormal bases. Projections when an orthonormal
basis is given. Orthogonal matrices. GramSchmidt process.
Examples.
 Lecture 13: GramSchmidt process and QR decomposition.
Existence and uniqueness. Connections with left inverses and
least squares. Examples.
 Lecture 14: Abstract inner product spaces and examples.
Abstract GrammSchmidt for spaces of functions. Legendre polynomials.
Fourier coefficients.
 Lecture 15: Determinants. Definition and properties. Computing
determinants via row reduction, via cofactors, via permutation matrices.
Invertible matrices.
 Lecture 16: Applications of determinants: finding inverses via
cofactors, Cramer's rule, volumes.
 Lecture 17: Eigenvalues and eigenvectors. Examples.
Characteristic polynomials. 2x2 matrices. Multiplicity of eigenvalues.
Relation to trace
and determinants.
 Lecture 18: Similar matrices, diagonalizable matrices.
Simultaneously diagonalizable matrices. Similar matrices share the same
characteristic polynomial, eigenvalues, trace, determinant.
 Lecture 19: Powers of matrices and exponentials via
diagonalization. Difference
and differential equations. Midterm review.
 Lecture 20: Fibonacci sequences. Stability of difference and
differential equations. Markov matrices. PerronFrobenius. Google page
rank. Other
examples.
 Lecture 21: Symmetric matrices and diagonalization. Skew
symmetric matrices and stability of differential equations. Complex
vectors. Hermitian product and properties.
Hermitian transpose.
 Lecture 22: Hermitian matrices, skew Hermitian matrices,
unitary matrices, normal matrices. Properties of their eigenvalues and
eigenspaces. Examples.
 Lecture 23: Unitary similarity. Unitary triangularization.
Unitary diagonalization. Examples. Spectral theorem for Hermitian and
symmetric matrices.
 Lecture 24: Quadratic forms and their definiteness. Criteria
for definiteness. Positive decomposition of symmetric
matrices. Writing quadratic forms as sums of squares.
 Lecture 25: Singular value decomposition. Left and right
singular vectors. Singular values. Calculating the singular value
decomposition and examples.
 Lecture 26: Applications of singular value decomposition.
Reading off the 4 subspaces of a matrix, projections onto the four
subspaces of a matrix, pseudoinverse, least squares solutions of minimal
length.
 Lecture 27: More examples and course review.
