# MATH 304-8 SPRING 2018 SYLLABUS

## Contact Information

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.

## Textbook and Course Contents

``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:

• 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

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

• Geometry in Linear Algebra: dot product, angles and lengths, orthogonality, orthonormal sets, orthogonal projections
• Orthogonal matrices
• 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

## Quizzes

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.

## Exams

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, 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 Rmn 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 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]. 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 Rn 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 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. 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 - 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. 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 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.

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.

## Homework

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.

### 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)