MATH 304 Fall 2018 SYLLABUS


``Linear Algebra, A Text for Math 304 Fall 2018 at Binghamton University'' by Matthew Brin and Gerald Marchesi, 12th edition, published by the Department of Mathematical Sciences, Binghamton University, and available through 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 may be covered is given at the end of this page.

Exams, Quizzes and Point Values

In each section the instructor will administer quizzes. There will be three common exams, each 90 minutes long, administered in the evening to all sections simultaneously. There will be one common Final Exam (2 hours long) during the scheduled Finals period. The quizzes will have a total point value of 75 points, the exams will be worth 75 points each, and the Final Exam will be worth 200 points. The material being tested in each exam will be determined and announced approximately one week before the exam. The Final Exam will be comprehensive, covering the whole course. ANYONE UNABLE TO TAKE AN EXAM SHOULD CONTACT THE INSTRUCTOR 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!

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

After each Exam is graded and returned, solutions will be posted on the main Math 304 webpage, along with a letter grade interpretation of the numerical score.

Any student with a special problem or a finals conflict must contact the instructor as soon as possible to make arrangements.

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

Determination of Letter Grades

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

General Comments

Class attendance is required at all scheduled meetings, and sleeping in class does not count as being there. Questions are welcomed at any time during a lecture. At the start of each class be ready to ask questions about homework problems or about the previous lecture. We want to create an atmosphere where you all feel very free to ask questions and make comments. If anyone feels that an instructor has not answered a question clearly, completely, and with respect and consideration for the student who asked it, please let your instructor know about it immediately so he/she can correct the problem. You can do this in class or in office hours, verbally or in writing, on paper or by email, or by whatever means makes you most comfortable, but with enough detail that your instructor can understand what you think was done wrong. It will be too late to help if you only complain at the end of the course. If you are not satisfied by the response of your instructor, please contact the course coordinator, Prof. Alex Feingold.

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 we 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 our knowledge of the subject, but to give you the depth of understanding expected of an adult with a university education in this subject. Some of your instructors have many 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 this advice, are not likely to be rewarded at the end. We are here to help and guide you, and we also grade the exams to judge how much you have learned, but grades are earned by you, not given by us. Exams will be a combination of theory questions and calculations appropriate for a course of this level.


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 may be considered as a factor in determining your course grade if you are a borderline case in the letter grade interpretation of the Total score.

Homework Assignments From the 12th Edition of this textbook.

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)

Course Contents

  • Systems of Linear Equations
  • Solution by row reduction
  • Matrices and operations with them
  • Reduced Row Echelon Form
  • Sets of Matrices of size mxn
  • Functions (general theory), injective, surjective, bijective, invertible, composition
  • Linear Functions determined by a matrix
  • Abstract Vector spaces
  • Basic theorems, examples, subspaces, linear combinations, span of a set of vectors
  • Linear functions L : V ---> W between vector spaces
  • Kernel and Range of a linear function
  • Connection with injective, surjective, bijective, invertible, composition of linear functions, isomorphism
  • Matrix multiplication from composition, formulas, properties (associativity)
  • Row (and column) operations achieved by left (right) matrix multiplication by elementary matrices
  • Theorems about invertibility of a square matrix (if row reduces to the identity matrix), algorithm to compute inverse
  • Linear independence/dependence, removing redundant vectors from a list keeping span the same
  • Basis (independent spanning set), Theorems about basis
  • Dimension of a vector space (or subspace), rank, nullity, theorems about dimension
  • For L : V ---> W, dim(V) = dim(Ker(L)) + dim(Range(L)) and its applications
  • Coordinates as an isomorphism from V to nx1 matrices
  • Representing a linear function L : V ---> W by a matrix (with respect to choice of basis S of V and basis T of W)
  • Theorems and algorithms, how matrix representing L changes when bases change to S' and T'
  • Equivalence of matrices, Block Identity Form
  • Study special case of L : V ---> V using same basis S on both ends, linear operators
  • Effect of change of basis on matrix representing a linear operator, similarity of matrices
  • Investigate when L might be represented by a diagonal matrix
  • Eigenvectors, Eigenvalues
  • Determinants as a tool for finding eigenvalues, general theorems and properties about determinants
  • det(AB) = det(A) det(B), A invertible iff det(A) not zero
  • Characteristic polynomial of a matrix, det(A - t I), roots are eigenvalues
  • Similar matrices have same characteristic polynomial
  • Geometric and algebraic multiplicities of eigenvalues for L : V ---> V (or for matrix A representing L)
  • Theorems (geometric mult less than or equal to algebraic mult), L diagonalizable iff geom mult = alg mult for all eigenvalues
  • Computational techniques to find a basis of eigenvectors, diagonalization of matrix A, if possible
  • Geometry in Linear Algebra: dot product, angles and lengths, orthogonality, orthonormal sets, orthogonal projections
  • Orthogonal matrices

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

  • Quadratic forms and associated bilinear forms on a vector space V
  • Matrix representing a bilinear form with respect to a choice of basis S
  • Effect of change of basis on matrix representing a bilinear form
  • Classification of quadratic forms

    Topics which may be covered on each exam

    These paragraphs will be updated before each exam is given.

    Topics which may be covered on Exam 1

    Here is a list of topics covered in lectures 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, Rmn, 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 LA: Rn --> Rm.

    Properties of general functions: one-to-one (injective), onto (surjective), both (bijective), invertible.

    Connection between properties of matrix A and function LA.

    Abstract definition of a real vector space, V. Examples, Rmn 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 LA composed with LB equals LAB. Lemma that LA = LB iff A=B.

    Defintion of standard basis vectors e1, ... , en in Rn and lemma that Aej = Colj(A), so AX is the sum of xj Colj(A).

    Definition of positive powers of a square matrix A, and positive powers of L: V ---> V. Definition of transpose of a matrix.

    Definition and some examples of subspaces.

    Topics which may be covered on Exam 2

    Topics that have been covered since Exam 1 and which may appear on Exam 2 are listed below.

    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 | In].

    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 S of V to give coordinates with respect to S for each vector v in V. How that coordinate function, [v]S, is a linear map from V to Rn when a basis S for V consists of n vectors.

    Transition matrices which give the relationship between the coordinates of a vector v with respect to different bases. If S and T are two bases of the same vector space, V, then the transition matrix from S to T is the square invertible matrix TPS such that [v]T = TPS [v]S.

    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 Rmn examples.

    Row-space and Column-space of a matrix, and their dimension related to the rank of the matrix.

    Information about the linear transformation LA: Rn--> Rm associated with rank(A).

    The relationship between the dimensions of Ker(L), Range(L) and V for L:V---> W.

    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.

    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)].

    If S and S' are two bases of V, and T and T' are two bases of W, and L:V---> W then there is a relationship between T[L]S, the matrix representing L from S to T, and T'[L]S', the matrix representing L from S' to T'.

    That relationship is T'[L]S' = T'QT T[L]S SPS' where SPS' is the transition matrix from S' to S, and T'QT is the transition matrix from T to T'.

    The concept of isomorphism (bijective linear map) and its properties.

    Topics which may be covered on Exam 3

    Topics that have been covered since Exam 2 and which may appear on Exam 3 are listed below.

    Chapter 5 material on determinants, their definition using permutations or by cofactor expansion, their properties, and methods of calculating them (definition by permutations or by cofactor expansions, crosshatching method for matrices of size n = 2 or n = 3 ONLY, using row operations).

    The use of determinant to get the characteristic polynomial, det(A - tIn), 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. Theorems about eigenspaces and diagonalizability.

    Independence of the union of bases for distinct eigenspaces.

    Geometric multiplicity and its relationship to algebraic multiplicity for each eigenvalue.

    Additional Topics which may be covered on the comprehensive Final Exam

    Additional topics that could be on the comprehensive Final Exam, include the following:

    Chapter 6 material on the dot product in Rn, 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.

    This page last modified on 11-10-2018.