Office: WH-115, Phone: 777-2465, Email: alex@math.binghamton.edu, Office Hours: MWF 1:10 - 2:10 and by appointment.

Class meets: MWF 4:40 - 6:10 in LN - 2403.

``Linear Algebra, A Text for Math 304 Spring 2018 at Binghamton University'' by Matthew Brin and Gerald Marchesi, 12th edition, published by the Department of Mathematical Sciences, Binghamton University, and available throught the University Bookstore.

Almost the entire book, Chapters 1 - 6, will be covered, parts of Chapter 7 if time permits. A list of major topics which will be covered is as follows:

Extra topics if time allows (usually not enough time for these in the elementary linear course) :

After each quiz is graded and returned, solutions will be posted here as pdf files.

Here is the link to Quiz 1 and its solutions.

Here is the link to Quiz 2 and its solutions.

Here is the link to Quiz 3 and its solutions.

Here is the link to Quiz 4 and its solutions.

Here is the link to Quiz 5 and its solutions.

Here is the link to Quiz 6 and its solutions.

Here is the link to Quiz 7 and its solutions.

Here is the link to Quiz 8 and its solutions.

Here is the link to Quiz 9 and its solutions.

Here is the link to Quiz 10 and its solutions.

Here is the link to Quiz 11 and its solutions.

Information about the scheduling of exams is posted on the main Math 304 webpage.

After each Exam is graded and returned, solutions will be posted here as pdf files.

Here is the link to Exam 1 and its solutions.

Here is the link to Exam 2 and its solutions.

Here is the link to a practice Exam 3 covering determinants and diagonalization: Practice Exam 3 and its solutions.

Here is the link to Exam 3 and its solutions.

Here is the link to a practice Final Exam covering everything: Practice Final Exam and its solutions.

Anyone with a special problem or a finals conflict must contact the professor as soon as possible to make arrangements. The Final Exam is comprehensive, covering the whole course.

There will be quizzes, 3 ``hourly'' exams (actually 1 hour and 30 minutes) and 1 Final Exam (2 hours long) during the scheduled Finals period. The quizzes will have a total point value of 75 points, the hourlies will be worth 75 points each, and the (2-hour) common Final Exam for all sections will be worth 200 points. The contents of each exam will be determined one week before the exam. The Final Exam will be comprehensive, covering the whole course. ANYONE UNABLE TO TAKE AN EXAM SHOULD CONTACT THE PROFESSOR AHEAD OF TIME TO EXPLAIN THE REASON. A MESSAGE CAN BE LEFT AT THE MATH DEPT OFFICE (777-2147) OR ON PROFESSOR FEINGOLD'S VOICEMAIL (777-2465). NO ONE SHOULD MISS THE FINAL!

There are links to practice problems of various kinds on Prof. Mazur's Math 304 webpage.

Here is a list of topics covered in lectures up to and including
which may be covered on Exam 1. Linear Systems, solving by
row reduction of the augmented matrix [A|B] and interpretation in
terms of free and dependent variables. Consistent vs. inconsistent
systems. Homogeneous systems AX=O. Elementary row operations and
reduction to Reduced Row Echelon Form (RREF). Matrices, the set of
all mxn real matrices, R^{m}_{n}, addition of
matrices, multiplication of a matrix by a real number (scalar).
Matrix shapes, names of special patterns. Rank of a matrix. How an
mxn matrix A determined a function L_{A}: R^{n}
--> R^{m}. Properties of general functions: one-to-one
(injective), onto (surjective), both (bijective), invertible.
Connection between properties of matrix A and function L_{A}
. Abstract definition of a real vector space, V. Examples, R^{m}_{n}
is a vector space. For any set S, the set F = {f : S ---> R} of
all functions from S to the reals R, is a vector space. Definition
of a linear transformation L : V ---> W from a vector space to a
vector space. Ker(L), Range(L) = Im(L). Basic facts about vector
spaces and about linear transformations, and examples. Defintion of
matrix multiplication AB through the definition L_{A}
composed with L_{B} equals L_{AB}. Lemma that L_{A}
= L_{B} iff A=B. Defintion of standard basis vectors e_{1},
... , e_{n} in R^{m}_{n} and lemma that Ae_{j}
= Col_{j}_{}(A), so AX is the sum of x_{j}
Col_{j}(A). Definition of positive powers of a
square matrix A, and positive powers of L: V ---> V. Definition
of transpose of a matrix. Definition of when a square matrix is
invertible, uniqueness of the inverse when it exists, and an
algorithm to decide and find it by row reduction of [A | I_{n}].
Definition and some examples of subspaces.

Topics that have been covered since Exam 1 and which may appear on Exam 2 are listed
below (including material covered in lectures up to and including ).
Elementary matrices and how they can be used to achieve elementary row or column operations.
The rules of matrix algebra, summarized on page 89 of the textbook.
The span of a set of vectors S in a vector space V, and why it forms a subspace of V.
How to check that a subset W in V is a subspace of V. Linear indepdendence or dependence of
a subset of V, definition and method of determining that. Theorems and examples about spanning
and independence, connection with rank of a matrix. Definition of a basis for a vector space,
and how to decide if a subset is a basis of V. Finding a basis for important examples of
subspaces, Ker(L), Range(L), where L:V---> W is a linear map. Use of a basis of V to give
coordinates for each vector in V, how that coordinate function is a linear map from V to
R^{n} when a basis for V consists of n vectors. Dimension of V as the number of
vectors in any basis for V. The standard basis for several examples of vector spaces, including
all the R^{m}_{n} examples. Row-space and Column-space of a matrix, and their
dimension related to the rank of the matrix. Information about the linear transformation
L_{A}: R^{n}--> R^{m} associated with rank(A). The relationship
between the dimensions of Ker(L), Range(L) and V for L:V---> W. Affine subsets of a vector
space related to the solutions of linear systems AX=B. Concepts about general vector spaces
given in Chapter 4, but already discussed in lectures. Extending an independent set to a basis,
cutting down a spanning set to a basis.
I hope this list will give you some idea of what will be covered on Exam 2.

Topics that have been covered since Exam 2 and which may appear on Exam 3 are listed below
(including material covered in lectures up to and including).
How to represent a general linear map L:V---> W with
respect to a choice of basis S in V and basis T in W by a matrix, that is, using coordinates
with respect to S, [ . ]_{S}, and coordinates with respect to T, [ . ]_{T}, to find
a matrix _{T}[L]_{S}, such that _{T}[L]_{S} [v]_{S} =
[L(v)]_{T}. The algorithm for finding that matrix by a row reduction of [T | L(S)].
The concept of isomorphism (bijective linear map) and its properties.
Chapter 5 material on determinants, their definition using permutations, their properties,
and methods of calculating them (definition, cofactor expansions, using row operations).
The use of determinant to get the characteristic polynomial, det(A - tI_{n}), whose roots
give the eigenvalues of A, and whose expression as a product of powers of distinct linear factors
gives the algebraic multiplicities. Eigenspaces, their properties, and how to decide if a matrix can
be diagonalized or not. Independence of the union of bases for distinct eigenspaces. Geometric multiplicity
and its relationship to algebraic multiplicity for each eigenvalue.

Additional topics that could be on the comprehensive Final Exam, include the following:
Chapter 6 material on the dot product in R^{n}, its properties, how it defines length of a vector
and orthogonality of vectors. Definition of the orthogonal complement of a set. Definition of a set of
vectors being orthogonal, or being orthonormal, and the connection with matrix multiplication and
transpose.
Orthogonal projection and the Gram-Schmidt orthogonalization process.

Each exam will be curved, giving each student a letter grade as well as a number grade, and the Total of all points earned will also be curved. The letter grades on the exams indicate how a student is doing, and will be taken into consideration in making the curve for the Totals. The course grade will be determined by the curve of Total points earned. Only borderline cases will be subject to further adjustment based on Homework. Any cases of cheating will be subject to investigation by the Academic Honesty Committee of Harpur College.

For each section of material covered there will be an assignment of problems from the textbook. Homework will be handled through the online system WeBWorK, and information about how to access it will be provided on the main Math 304 webpage. The number of homeworks attempted will be considered as a factor in determining your course grade if you are a borderline case in the Total curve.

Exercises (1) - (4)

Exercises (5)-(6)

Exercises (7)

Exercises (8)

Exercises (9)

Exercises (10)

Exercises (11)

Exercises (12)-(13)

Exercises (14)-(18)

Exercises (19), (20), (21), (22), (23), (25)

Exercises (24), (26)

Exercises (27), (28), (29)

Exercises (34), (35), (36), (37), (38)

Exercises (39), (40)

Exercises (41), (42), (49), (50), (S3-18), (S3-19)

Exercises (43), (45), (46)

Exercises (48)

Exercises (47), (51), (52), (53), (54)

Exercises (55)

Exercises (56,a-e)

Exercises (58)

Exercises (59), (60)

Exercises (61), (62)

Exercises (64)

Exercises (65)

Exercises (66), (67)

Exercises (68), (69)

Class attendance is required at all scheduled meetings, and sleeping in class does not count as being there. Lectures can be interrupted at any time for questions. At the start of each class be ready to ask questions about homework problems or about the previous lecture. I want to create an atmosphere where you all feel very free to ask questions and make comments. If anyone feels that I have not answered a question clearly, completely, and with respect and consideration for the student who asked it, I want you to let me know about it immediately so I can correct the problem. You can do this in class or in my office hours, verbally or in writing, on paper or by email, or by whatever means makes you most comfortable, but with enough detail that I understand what you think I did wrong. It will be too late to help if you only tell me at the end of the course.

The material is a combination of theory and calculation, and it is necessary to understand the theory in order to do sensible calculations and interpret them correctly. There is a significant difference between training and education, and I feel strongly that our goal at this university is to educate you, not just to train you to do computations. Theory is not presented to impress you with my knowledge of the subject, but to give you the depth of understanding expected of an adult with a university education in this subject. I will try to give you the benefit of my 37 years of experience teaching mathematics at the university level, but it will require your consistent concentrated study to master this material. While much learning can take place in the classroom, a significant part of it must be done by you outside of class. Using the book, class notes, homework exercises, only you can achieve success in this course. Students who do not take this course seriously, who do not take the advice I give, are not likely to be rewarded at the end. I am here to help and guide you, and I also make and grade the exams to judge how much you have learned, but grades are earned by you, not given by me. Exams will be a combination of theory questions and calculations appropriate for a course of this level.

This page last modified on 5-7-2018.