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Course: MATH 367, Spring 2011
School: Middle East Technical...
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0 Prerequisites All Chapter topics listed in this chapter are covered in A Primer of Abstract Mathematics by Robert B. Ash, MAA 1998. 0.1 Elementary Number Theory The greatest common divisor of two integers can be found by the Euclidean algorithm, which is reviewed in the exercises in Section 2.5. Among the important consequences of the algorithm are the following three results. 0.1.1 If d is the greatest...

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0 Prerequisites All Chapter topics listed in this chapter are covered in A Primer of Abstract Mathematics by Robert B. Ash, MAA 1998. 0.1 Elementary Number Theory The greatest common divisor of two integers can be found by the Euclidean algorithm, which is reviewed in the exercises in Section 2.5. Among the important consequences of the algorithm are the following three results. 0.1.1 If d is the greatest common divisor of a and b, then there are integers s and t such that sa + tb = d. In particular, if a and b are relatively prime, there are integers s and t such that sa + tb = 1. 0.1.2 If a prime p divides a product a1 an of integers, then p divides at least one ai 0.1.3 Unique Factorization Theorem If a is an integer, not 0 or 1, then (1) a can be written as a product p1 pn of primes. (2) If a = p1 pn = q1 qm , where the pi and qj are prime, then n = m and, after renumbering, pi = qi for all i. [We allow negative primes, so that, for example, 17 is prime. This is consistent with the general denition of prime element in an integral domain; see Section 2.6.] 1 2 0.1.4 CHAPTER 0. PREREQUISITES The Integers Modulo m If a and b are integers and m is a positive integer 2, we write a b mod m, and say that a is congruent to b modulo m, if a b is divisible by m. Congruence modulo m is an equivalence relation, and the resulting equivalence classes are called residue classes mod m. Residue classes can be added, subtracted and multiplied consistently by choosing a representative from each class, performing the appropriate operation, and calculating the residue class of the result. The collection Zm of residue classes mod m forms a commutative ring under addition and multiplication. Zm is a eld if and only if m is prime. (The general denitions of ring, integral domain and eld are given in Section 2.1.) 0.1.5 (1) The integer a is relatively prime to m if and only if a is a unit mod m, that is, a has a multiplicative inverse mod m. (2) If c divides ab and a and c are relatively prime, then c divides b. (3) If a and b are relatively prime to m, then ab is relatively prime to m. (4) If ax ay mod m and a is relatively prime to m, then x y mod m. (5) If d = gcd(a, b), the greatest common divisor of a and b, then a/d and b/d are relatively prime. (6) If ax ay mod m and d = gcd(a, m), then x y mod m/d. (7) If ai divides b for i = 1, . . . , r, and ai and aj are relatively prime whenever i = j , then the product a1 ar divides b. (8) The product of two integers is their greatest common divisor times their least common multiple. 0.1.6 Chinese Remainder Theorem If m1 , . . . , mr are relatively prime in pairs, then the system of simultaneous equations x bj mod mj , j = 1, . . . , r, has a solution for arbitrary integers bj . The set of solutions forms a single residue class mod m=m1 mr , so that there is a unique solution mod m. This result can be derived from the abstract form of the Chinese remainder theorem; see Section 2.3. 0.1.7 Eulers Theorem The Euler phi function is dened by (n) = the number of integers in {1, . . . , n} that are relatively prime to n. For an explicit formula for (n), see Section 1.1, Problem 13. Eulers theorem states that if n 2 and a is relatively prime to n, then a(n) 1 mod n. 0.1.8 Fermats Little Theorem If a is any integer and p is a prime not dividing a, then ap1 1 mod p. Thus for any integer a and prime p, whether or not p divides a, we have ap a mod p. For proofs of (0.1.7) and (0.1.8), see (1.3.4). 0.2. SET THEORY 0.2 3 Set Theory 0.2.1 A partial ordering on a set S is a relation on S that is reexive (x x for all x S ), antisymmetric (x y and y x implies x = y ), and transitive (x y and y z implies x z ). If for all x, y S , either x y or y x, the ordering is total. 0.2.2 A well-ordering on S is a partial ordering such that every nonempty subset A of S has a smallest element a. (Thus a b for every b A). 0.2.3 Well-Ordering Principle Every set can be well-ordered. 0.2.4 Maximum Principle If T is any chain (totally ordered subset) of a partially ordered set S , then T is contained in a maximal chain M . (Maximal means that M is not properly contained in a larger chain.) 0.2.5 Zorns Lemma If S is a nonempty partially ordered set such that every chain of has S an upper bound in S , then S has a maximal element. (The element x is an upper bound of the set A if a x for every a A. Note that x need not belong to A, but in the statement of Zorns lemma, we require that if A is a chain of S , then A has an upper bound that actually belongs to S .) 0.2.6 Axiom of Choice Given any family of nonempty sets Si , i I , we can choose an element of each Si . Formally, there is a function f whose domain is I such that f (i) Si for all i I . The well-ordering principle, the maximum principle, Zorns lemma, and the axiom of choice are equivalent in the sense that if any one of these statements is added to the basic axioms of set theory, all the others can be proved. The statements themselves cannot be proved from the basic axioms. Constructivist mathematics rejects the axiom of choice and its equivalents. In this philosophy, an assertion that we can choose an element from each Si must be accompanied by an explicit algorithm. The idea is appealing, but its acceptance results in large areas of interesting and useful mathematics being tossed onto the scrap heap. So at present, the mathematical mainstream embraces the axiom of choice, Zorns lemma et al. 4 0.2.7 CHAPTER 0. PREREQUISITES Proof by Transnite Induction To prove that statement Pi holds for all i in the well-ordered set I , we do the following: 1. Prove the basis step P0 , where 0 is the smallest element of I . 2. If i > 0 and we assume that Pj holds for all j < i (the transnite induction hypothesis), prove Pi . It follows that Pi is true for all i. 0.2.8 We say that the size of the set A is less than or equal to the size of B (notation A s B ) if there is an injective map from A to B . We say that A and B have the same size (A =s B ) if there is a bijection between A and B . 0.2.9 Schrder-Bernstein Theorem o If A s B and B s A, then A =s B . (This can be proved without the axiom of choice.) 0.2.10 Using (0.2.9), one can show that if sets of the same size are called equivalent, then s on equivalence classes is a partial ordering. It follows with the aid of Zorns lemma that the ordering is total. The equivalence class of a set A, written |A|, is called the cardinal number or cardinality of A. In practice, we usually identify |A| with any convenient member of the equivalence class, such as A itself. 0.2.11 For any set A, we can always produce a set of greater cardinality, namely the power set 2A , that is, the collection of all subsets of A. 0.2.12 Dene addition and multiplication of cardinal numbers by |A| + |B | = |A B | and |A||B | = |A B |. In dening addition, we assume that A and B are disjoint. (They can always be disjointized by replacing a A by (a, 0) and b B by (b, 1).) 0.2.13 If 0 is the cardinal number of a countably innite set, then 0 + 0 = 0 0 = 0 . More generally, (a) If and are cardinals, with and innite, then + = . (b) If = 0 (i.e., is nonempty), and is innite, then = . 0.2.14 If A is an innite set, then A and the set of all nite subsets of A have the same cardinality. 0.3. LINEAR ALGEBRA 0.3 5 Linear Algebra It is not feasible to list all results presented in an undergraduate course in linear algebra. Instead, here is a list of topics that are covered in a typical course. 1. Sums, products, transposes, inverses of matrices; symmetric matrices. 2. Elementary row and column operations; reduction to echelon form. 3. Determinants: evaluation by Laplace expansion and Cramers rule. 4. Vector spaces over a eld; subspaces, linear independence and bases. 5. Rank of a matrix; homogeneous and nonhomogeneous linear equations. 6. Null space and range of a matrix; the dimension theorem. 7. Linear transformations and their representation by matrices. 8. Coordinates and matrices under change of basis. 9. Inner product spaces and the projection theorem. 10. Eigenvalues and eigenvectors; diagonalization of matrices with distinct eigenvalues, symmetric and Hermitian matrices. 11. Quadratic forms. A more advanced course might cover the following topics: 12. Generalized eigenvectors and the Jordan canonical form. 13. The minimal and characteristic polynomials of a matrix; Cayley-Hamilton theorem. 14. The adjoint of a linear operator. 15. Projection operators. 16. Normal operators and the spectral theorem.
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Middle East Technical University - MATH - 367
Chapter 1Group Fundamentals1.11.1.1Groups and SubgroupsDenitionA group is a nonempty set G on which there is dened a binary operation (a, b) absatisfying the following properties.Closure : If a and b belong to G, then ab is also in G;Associativit
Middle East Technical University - MATH - 367
Chapter 2Ring Fundamentals2.12.1.1Basic Denitions and PropertiesDenitions and CommentsA ring R is an abelian group with a multiplication operation (a, b) ab that is associativeand satises the distributive laws: a(b + c) = ab + ac and (a + b)c = ab
Middle East Technical University - MATH - 367
Chapter 3Field Fundamentals3.1Field ExtensionsIf F is a eld and F [X ] is the set of all polynomials over F, that is, polynomials withcoecients in F , we know that F [X ] is a Euclidean domain, and therefore a principal idealdomain and a unique fact
Middle East Technical University - MATH - 367
Chapter 4Module Fundamentals4.1Modules and Algebras4.1.1Denitions and CommentsA vector space M over a eld R is a set of objects called vectors, which can be added,subtracted and multiplied by scalars (members of the underlying eld). Thus M is anab
Middle East Technical University - MATH - 367
Chapter 5Some Basic Techniques ofGroup Theory5.1Groups Acting on SetsIn this chapter we are going to analyze and classify groups, and, if possible, break downcomplicated groups into simpler components. To motivate the topic of this section, letsloo
Middle East Technical University - MATH - 367
Chapter 6Galois Theory6.1Fixed Fields and Galois GroupsGalois theory is based on a remarkable correspondence between subgroups of the Galoisgroup of an extension E/F and intermediate elds between E and F . In this sectionwe will set up the machinery
Middle East Technical University - MATH - 367
Chapter 7Introducing Algebraic NumberTheory(Commutative Algebra 1)The general theory of commutative rings is known as commutative algebra. The mainapplications of this discipline are to algebraic number theory, to be discussed in thischapter, and al
Middle East Technical University - MATH - 367
Chapter 8Introducing AlgebraicGeometry(Commutative Algebra 2)We will develop enough geometry to allow an appreciation of the Hilbert Nullstellensatz,and look at some techniques of commutative algebra that have geometric signicance. Asin Chapter 7, u
Middle East Technical University - MATH - 367
Chapter 9Introducing NoncommutativeAlgebraWe will discuss noncommutative rings and their modules, concentrating on two fundamental results, the Wedderburn structure theorem and Maschkes theorem. Further insightinto the structure of rings will be provi
Middle East Technical University - MATH - 367
Chapter 10Introducing HomologicalAlgebraRoughly speaking, homological algebra consists of (A) that part of algebra that is fundamental in building the foundations of algebraic topology, and (B) areas that arise naturallyin studying (A).10.1Categorie
Middle East Technical University - MATH - 367
BIBLIOGRAPHYGeneralCohn, P.M., Algebra, Volumes 1 and 2, John Wiley and Sons, New York, 1989Dummit, D.S. and Foote, R.M., Abstract Algebra, Prentice-Hall, Upper Saddle River,NJ, 1999Hungerford, T.M., Algebra, Springer-Verlag, New York, 1974Isaacs, I
Middle East Technical University - MATH - 367
EnrichmentChapters 14 form an idealized undergraduate course, written in the style of a graduatetext. To help those seeing abstract algebra for the rst time, I have prepared this section,which contains advice, explanations and additional examples for e
Middle East Technical University - MATH - 367
Abstract Algebra: The Basic Graduate YearRobert B. AshPREFACEThis is a text for the basic graduate sequence in abstract algebra, oered by mostuniversities. We study fundamental algebraic structures, namely groups, rings, elds andmodules, and maps bet
Middle East Technical University - MATH - 367
Solutions Chapters 15Section 1.11. Under multiplication, the positive integers form a monoid but not a group, and thepositive even integers form a semigroup but not a monoid.2. With |a| denoting the order of a, we have |0| = 1, |1| = 6, |2| = 3, |3| =
Middle East Technical University - MATH - 367
Solutions Chapters 610Section 6.11. We have r1 = 2, r2 = 1, r3 = 1 so t1 = 1, t2 = 0, t3 = 1. The algorithm terminates inone step after after subtraction of (X1 + X2 + X3 )(X1 X2 X3 ). The given polynomialcan be expressed as e1 e3 .2. We have r1 = 2,
Middle East Technical University - MATH - 367
Supplement: The Long ExactHomology Sequence andApplicationsS1. Chain ComplexesIn the supplement, we will develop some of the building blocks for algebraic topology.As we go along, we will make brief comments [in brackets] indicating the connectionbe
Middle East Technical University - MATH - 367
Math 420, Point Set TopologySuggested exercises 1Problem 1. Consider the following collection of subsets of RB = cfw_B : R\Bis nite(a) Show that B is a basis for topology on R.(b) Determined if the generated topological space is Hausdor.Problem 2.
Middle East Technical University - MATH - 367
Math 420, Point Set TopologySuggested exercises 2Problem 1. Let be the uniform metric on R . Given x = cfw_xn R and (0, 1), letU (x, ) = (x1 + , x1 ) (xn + , xn ) . . .a. Show that U (x, ) is not equal to the -ball B (x, ).b. Show that U (x, ) is n
Middle East Technical University - MATH - 367
Math 420, Point Set TopologySuggested exercisesProblem 1. Let I and I be two topologies on X . Show that if X is acompact Hausdor space under both I and I , then either I = I or theyare not comparable.Problem 2. Is the interval [0, 1] compact in the
Middle East Technical University - MATH - 367
Abstract Algebra done ConcretelyDonu ArapuraFebruary 19, 2004Introduction: I wrote these notes for my math 453 class since I couldntnd a book that covered basic abstract algebra with the level and emphasis thatI wanted. Rather than spending a lot of
Middle East Technical University - MATH - 367
Elementary Abstract AlgebraW. Edwin ClarkDepartment of MathematicsUniversity of South Florida(Last revised: December 23, 2001)Copyright c 1998 by W. Edwin ClarkAll rights reserved.iiiPrefaceThis book is intended for a one semester introduction t
Middle East Technical University - MATH - 367
Incomplete Notes onFraleighs Abstract Algebra (4th ed.)Afra ZomorodianDecember 12, 2001Contents1 A Few Preliminaries1.1 Mathematics and Proofs . . . . .1.2 Sets and Equivalence Relations .1.3 Mathematical Induction . . . . .1.4 Complex and Matrix
Middle East Technical University - MATH - 367
ABSTRACT ALGEBRA:A STUDY GUIDEFOR BEGINNERSJohn A. BeachyNorthern Illinois University20062This is a supplement toAbstract Algebra, Third Editionby John A. Beachy and William D. BlairISBN 1577664344, Copyright 2005Waveland Press, Inc.4180 IL Ro
Middle East Technical University - MATH - 367
ABSTRACT ALGEBRA:A STUDY GUIDEFOR BEGINNERSJohn A. BeachyNorthern Illinois University2000iiThis is a supplement toAbstract Algebra, Second Editionby John A. Beachy and William D. BlairISBN 0881338664, Copyright 1996Waveland Press, Inc.P.O. Box
Middle East Technical University - MATH - 367
Jonathan BergknoHerstein SolutionsChapters 1 and 2Throughout, G is a group and p is a prime integer unless otherwise stated. A B denotes that A is asubgroup of B while AB denotes that A is a normal subgroup of B .H 1.3.14* (Fermats Little Theorem) P
Middle East Technical University - MATH - 480
G. Evans, J. Blackledge and P. YardleyNumerical Methodsfor Partial DifferentialEquationsWith 55 FiguresSpringerContents1.B ackground Mathematics11.1 Introduction11.2 Vector and Matrix Norms31.3 Gerschgorin's Theorems71.4 Iterative Solution
Middle East Technical University - MATH - 480
Math 480Spring 2011Homework IDue date, March 14th, 2011Chapter 3: The Taylor series methodChapter 4: Linear Multistep Methods-I: Construction and ConsistencyChapter 5 Linear Multistep MethodsII: Convergence and Zero-Stability1. pp. 41, Exercise 3.8
Middle East Technical University - MATH - 480
G1BINMIntroduction to Numerical Methods717 Iterative methods for matrixequations7.1The need for iterative methodsWe have seen that Gaussian elimination provides a method for nding the exact solution(if rounding errors can be avoided) of a system o
Middle East Technical University - MATH - 480
Crash course in MATLABc Tobin A. Driscoll. All rights reserved. June 23, 2006The purpose of this document is to give a medium-length introduction to the essentials of MATLAB and how to use it well. Im aiming for a document thats somewhere between a twop
Middle East Technical University - MATH - 480
An Introduction to MatlabVersion 2.3David F. GrithsDepartment of Mathematics The University Dundee DD1 4HNWith additional material by Ulf Carlsson Department of Vehicle Engineering KTH, Stockholm, SwedenCopyright c 1996 by David F. Griths. Amended Oc
Middle East Technical University - MATH - 480
Edward Neuman Department of Mathematics Southern Illinois University at Carbondale edneuman@siu.eduThis tutorial is intended for those who want to learn basics of MATLAB programming language. Even with a limited knowledge of this language a beginning
Middle East Technical University - MATH - 480
MATH480Numerical Solution of Differential EquationsMIDTERM EXAMApril, 27st,20111. (10 poinis) Given hat r is a positi'e real number. consider the jineartwo-step method'n_2x. -J 4,- : h &quot; * -Af .*f &quot;)1U&quot;-zQInDetermine the set of all cu such th
Middle East Technical University - MATH - 480
F1.4ZH2 Numerical Solution of PDEs1.4Page 7Taylor series and dierence operatorsWe can use Taylor series expansions in one of the variables to see how well Finite Dierence(FD) approximations work. Consider the approximation (u(x + x, t) u(x, t)/x to u
Middle East Technical University - MATH - 480
F1.4ZH2 Numerical Solution of PDEs2.6.1Page 23LTE analysis of the -methodThe PDE ut = uxx is approximated on a uniform grid of size x = 1/N in space and t intime, with approximate solutionnwj u(xj , tn ) ,xj = x0 + j x , tn = nt,where u(x, t) is
Middle East Technical University - MATH - 480
F1.4ZH2 Numerical Solution of PDEsPage 15LTEs (continued)The rst term on the right of the LTE for the FTCS scheme is referred to as the leading termof the local truncation error (LTE): for this scheme it is rst order accurate in time and 2ndorder in
Middle East Technical University - MATH - 373
Middle East Technical University - MATH - 373
Middle East Technical University - MATH - 373
Middle East Technical University - MATH - 373
Problems in Geometry (1)1. In a triangle ABC with orthocenter H and circumcircle O prove that&lt; BAH =&lt; OAC .12. Let ABC be a triangle with orthocenter H in which AH, BH, CH meet BC, CA, ABin D, E, F respectively. Prove thatHA HD = HB HE = HC HF.23.
Middle East Technical University - MATH - 373
Problems in Geometry (6)1. What is the locus of a point whereof the power with respect to a xed circle is aconstant ?2. Compute the power of the point P (x0 , y0 ) with respect to the circle x2 + y 2 +2ax +2by +c = 0. Write down the equation of the ra
Middle East Technical University - MATH - 373
Problems in Geometry (7)1. Let be an ellipse with foci F, F . If the tangent lines at A, B intersect in P ,prove that P F AB i F AB . 12. Let be an ellipse with foci F, F . Let t, t be the tangents to which intersect in P .If H, H are respectively the
Middle East Technical University - MATH - 373
Problems in Geometry (2)1. Given a rectangle ABCD prove that |P A|2 + |P C |2 = |P B |2 + |P D|2 for any point P.2. Given an acute angled positively oriented triangle ABC, let O and H be the circumcenter and the orthocenter of ABC, respectively. Let AH
Middle East Technical University - MATH - 373
Problems in Geometry (3)1. Given a triangle ABC, let A be the midpoint of [B, C ] and consider Y CA cfw_C, A,Z AB cfw_A, B such that BY and CZ meet on AA . Prove that Y Z is parallel to BC.12. Given a triangle ABC, consider P BC cfw_B, C , Q CA cfw_C,
Middle East Technical University - MATH - 373
Problems in Geometry (4)1. Consider a quadrangle ABCD with points P, Q, R, S, T on BD, AB, AD, CB, CDrespectively. Prove that if P, Q, R are collinear and P, S, T are collinear, then the lines1QS, RT, AC are concurrent.2. The following is a problem b
Middle East Technical University - MATH - 373
Problems in Geometry (5)1. Given a circle , consider points P, C, D and let [A, B ] be a diameter of . Provethat the Simson line of P with respect to the triangle ACD is perpendicular to the Simsonline of P with respect to the triangle BCD.12. Given a
Middle East Technical University - MATH - 373
Problems in Geometry (8)1. (A) What is the image of the line 2x + y = 2 under Tr[1,3] ?(B) What is the image of the line x = 1 under Rot(0, 0), /4) ?(C) What is the image of the line x + y = 1 under Refk , where k is the line y = 2x ?12. (A) Let P, Q
Middle East Technical University - MATH - 420
Fall 2003- First Midterm1)Let ( X , ) be a topological space and A a subset of X.(a) x Bdd ( A) if and only if.x Int ( A) if and only if.(b) Show that Bdd ( A) Int ( A) = .2) Prove or disprove:(a) = u : eithercfw_uorucisfinite is a topolgy on
Middle East Technical University - MATH - 420
Chapter 2Topological SpacesA topological space (X, ) is a set X with a topology , i.e., a collection ofsubsets of X with the following properties:1. X , .2. If A, B , then A B .3. For any collection cfw_A , if all A , then A .The sets in are calle
UC Irvine - BIO94 - 101
Lec 2 Evolution:Darwinian Fitness:Adaptation:Acclimation:Genetic Correlation:Fitness trade-off:Lec 3:4 Evolutionary Mechanism:Genetic Drift:Change in characteristics ofindividuals over time. It results fromnatural selection. It is the heritable
University of Texas - CHEMISTRY - 50960
galloway (jlg4528) H11: Thermo 1 mccord (50960)This print-out should have 15 questions.Multiple-choice questions may continue onthe next column or page nd all choicesbefore answering.This homework really comes from sections1 and 2 in Chapter 9. Also
University of Texas - CHEMISTRY - 50960
galloway (jlg4528) H10: IMFs mccord (50960)This print-out should have 25 questions.Multiple-choice questions may continue onthe next column or page nd all choicesbefore answering.IMFs are intermolecular forces.Temporary dipoles are also called insta
HCCS - PSYC - 2301
PLEASE NOTE: The Instructors Resources files lose their formattingin the conversion from Quark XPress to Microsoft Word. The finalformatted files are also available in Adobe PDF for your convenience.CH 2: NEUROSCIENCE AND BEHAVIORCHAPTER PREVIEWOur n
Guilford Tech - ENGLISH - 113
GWU - ECE - 3220
Joshua DeanLaboratory 01: Introduction to MATLABLab 02b: Introduction to Complex Exponentials- Direction FindingFebruary 5, 2011The George Washington UniversitySchool of Engineering and Applied ScienceECE 3220Design of Logic SystemsLab Section 30
GWU - ECE - 3220
Joshua DeanLab 03: AM and FM Sinusoidal SignalsFebruary 14, 2011The George Washington UniversitySchool of Engineering and Applied ScienceECE 3220Digital Signal ProcessingLab Section 30GTA: Damon Conover85/100* In the future, you should use the w
GWU - ECE - 3220
Joshua DeanLab 06: Digital Images: A/D and D/AFebruary 28, 2011The George Washington UniversitySchool of Engineering and Applied ScienceECE 3220Digital Signal ProcessingLab Section 30GTA: Damon Conover95/1003.1 Down-Sampling[25/25] Describe how
GWU - ECE - 3220
Joshua DeanLab 7: Sampling, Convolution, and FIR FilteringApril 11, 2011The George Washington UniversitySchool of Engineering and Applied ScienceECE 3220Digital Signal ProcessingLab Section 30GTA: Damon ConoverTotal Grade = 95/100[10] Section 3.
GWU - ECE - 3220
Joshua DeanLab 10: Octave Band FilteringApril 25, 2011The George Washington UniversitySchool of Engineering and Applied ScienceECE 3220Digital Signal ProcessingLab Section 30GTA: Damon ConoverTotal Grade = 90/100[10] Plot the frequency responses
GWU - ECE - 3220
Joshua DeanLab 15: Fourier SeriesMarch 28, 2011The George Washington UniversitySchool of Engineering and Applied ScienceECE 3220Digital Signal ProcessingLab Section 30GTA: Damon ConoverTotal Grade = 90/100[10] Plot magnitude and phase of H(jw) [
GWU - ECE - 3220
Joshua DeanECE 3220Prelab #3[-35] 7 days late[-5]60/100a) Determine the total duration of the synthesized signal in seconds, and also the length ofthe tt vector (number of samples).[-5] 1.8 secondsdur/dt = [1.8/(1/11025)]+1 = 19846s samplesb)c)
GWU - ECE - 3220
Joshua DeanECE 3220 Lab Section 30 Prelab 06 Digital Images: A/D and D/ADamon ConoverDue: 2/7/2011100/1001.4 Get Test Images(a) Load and display the 326 426 lighthouse image from lighthouse.mat.Use whos to check the size of ww after loading.EDU&gt; l