Instructor 
Dragos Oprea
 Office: APM 6101
 Office hour: Tuesdays 45pm,
Wednesdays 3:154:15. I am also availabe for questions
after lecture.
 Email: doprea at math dot ucsd dot
edu

Lectures: 
MWF, 12pm12:50pm, Peterson 102. 
Course Assistant 
Christian Woods
 Discussion: Thursday 1010:50AM or
Thursday 1111:50, APM
7421
 Office: APM 6436
 Office
Hours: Mondays 13pm and Thursdays 24pm.
 Email:
c1woods at
ucsd dot edu

Course Content 
This course is the first in the Mathematics Department's Honors
sequence.
It covers basic linear algebra: vectors and matrices, solution of systems
of linear equations, geometry of vector spaces, linear transformations,
eigenvalues and eigenvectors.
The three courses in the Honors sequence, Math 31ABC, cover
essentially
the same material as do Math 20F, 20C, and 20E, respectively, but at a
more sophisticated conceptual level. The Honors sequence emphasizes
proofs, so students completing it will be exempt from taking Math 109
(Mathematical Reasoning). The Honors sequence is intended for mathematics
majors and
prospective mathematics majors (although others are very welcome) and
gives a much better view of what upperdivision mathematics is like than
does the standard Math 20 sequence.

Prerequisities: 
AP calculus in high school,
with a 5 on the BC exam, or permission from the instructor. A grade of B
or better is necessary to continue from one course to the next in the
sequence. 
Grade Breakdown  The grade is computed as the
following
weighed average:
Homework 20%, Midterm I 20%, Midterm II
20%,
Final Exam 40%. 
Textbook:  J. H. Hubbard and B. B.
Hubbard, Vector
Calculus, Linear Algebra, and Differential Forms: A
Unified Approach, Fourth Edition. 
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 Fridays at 4:30PM
in
the TA's mailbox. The tentative dates for the homeworks are Oct 4, Oct
11,
Oct 18, Nov 1, Nov 8, Nov 15, Dec 4.
You may work together with your
classmates
on your
homework
and/or ask the TA (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
on October 21 and November 18. There will be no makeup
exams. 
Final Exam  The final examination will be held on
Thursday, December 12, 11:302:30, Peterson 102. 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 
 Friday, September 27: First lecture
 Monday, October 21: Midterm I
 Monday, November 11: Veterans' Day. No
class
 Monday, November 18: Midterm II
 ThursdayFriday, November 2829:
Thanksgiving. No class.
 Friday, December 6: Last Lecture
 Thursday, December 12: FINAL EXAM,
11:30AM2:30PM
pm

Exams 
 Preparation for Midterm 1:
 Preparation for Midterm 2:
 Preparation for Final Exam:

Lecture Summaries  
Lecture 1: Introduction. Vectors. Addition. Scalar
multiplication. Vector subspaces.
 Lecture 2: Matrices. Operations with matrices: addition,
scalar multiplication, multiplication, properties, matrixvector product,
inverses, transpose.
 Lecture 3: Dot product. Angle between vectors. The
CauchySchwartz inequality. The triangle inequality. Length of matrices.
 Lecture 4: Cross product and properties. Solving systems
of linear equations. Row operations. Row reduced echelon form.
 Lecture 5: Solving systems using row reduction. Pivot and free
variables. Uniqueness and existence of solutions. Existence of the
rowreduced echelon form. Mathematical induction.
 Lecture 6: Linear independence. Span. Basis. Dimension of a
subspace.
 Lecture 7: Inverses. Finding inverses by row
reduction. Elemenatary matrices.
 Lecture 8: Null space. Basis for the null space.
Dimension of the null space is given by the number of free variables.
Determining if the columns of a matrix are linearly dependent.
 Lecture 9: Column space. Finding the column space by
rowreducing the augumented matrix. Basis for column space
is given by the pivot columns. Rank. Ranknullity theorem.
 Lecture 10: Row space and rowrank. Rowrank equals
columnrank. Inverses and column space/null space. Left inverses and right
inverses. Midterm review.
 Lecture 11: Linear transformations. Matrix of a linear
transformation. Geometric examples: projections, reflections, rotations.
 Lecture 12: Injective, surjective, bijective maps. Image and
preimage. Reinterpretation of these notions for linear transformations.
Composition of linear transformations and matrix multiplication.
 Lecture 13: Determinants. Multilinear, antisymmetric,
normalized functions. Expansion along columns. Effect of column operations
on determinants.
 Lecture 14: Uniquness of determinants as
multilinear, antisymmetric, normalized functions. Invertible
matrices have nonzero determinants. Determinants
distributes over matrix products.
 Lecture 15: Connection between determinants and
volumes. Existance of determinants as multilinear, antisymmetric,
normalized functions. More on inductive proofs.
 Lecture 16: Every invertible
matrix is product of elementary matrices. Determinant of the transpose.
Determinants and row operations. Permutations and signature. Permutation matrices.
 Lecture 17: Determinants and
permutations. Orthogonal complements and properties.
 Lecture 18: Connections between the four subspaces of a
matrix. Matrix of the projection onto the column space. Connection
with left inverses.
 Lecture 19: Projections onto row space, column space, null
space, left null space. Least square solutions and connection with the
left inverse. Best fit line.
 Lecture 20: Orthogonal bases, orthogonal matrices. Formulas
for projections in the presence of an orthonormal basis. GramSchmidt
process.
 Lecture 21: Systems of coordinates. Change of basis matrix.
Matrix of a linear transformation with respect to an arbitrary basis.
Example: reflection in a plane in R^3.
 Lecture 22: Similar matrices. Matrices of linear
transformation in different bases are similar. Diagonalizable linear
transformations. Eigenvalues, eigenspaces, characteristic polynomial.
 Lecture 23: Examples of eigenvalues and eigenspaces. Sum and
product of
eigenvalues in terms of trace and determinant. Diagonalizable matrices in
terms of geometric/algebraic multiplicities.
 Lecture 24: Similar matrices have the same characteristic
polynomial. Computing powers of diagonalizable matrices. CayleyHamilton
theorem. Symmetric matrices and the spectral theorem.
 Lecture 25: More on the spectral theorem and examples.
Quadratic forms and symmetric matrices.
 Lecture 26: Signature of quadratic forms and classification in
terms of eigenvalues. Sylvester's criterion.
