Instructor: Dr. Eyal Z. Goren.Time: MTWThF 9:05-10:55 Location: BURN 1B23Syllabus (section numbers referto Poole's book): Geometric vectors, dot product, lines andplanes(Sections 1.1, 1.2, 1.3, Cross Products); Systems of linear equations,spanning sets and linear independence (Sections 2.1, 2.2, 2.3);Matrices,matrix algebra, subspaces, basis, dimension, rank, lineartransformations(Sections 3.1, 3.2, 3.3, 3.4, 3.5); Eigenvalues and eigenvectors,determinants,similarity and diagonalization (Sections 4.1, 4.2, 4.3, 4.4);Orthogonality,orthogonal complements and orthogonal projections, the Gram-Schmidtprocess.(Sections 5.1, 5.2, 5.3,5.4, 5.5(Quadratic Forms)).Prerequisite: A course infunctions.See Calendar for additional information.Text Book: Linear Algebra,by D. Poole.Additional Texts: The text bookwill suffice for everything in this course. You may however consultalsothe following texts if you need.1) Anton, H. "Elementary linear algebra",7th ed., or Anton & Rorres "Elementary linear algebra, applicationsversion", 8th ed., Chapters 1-7.2) Nicholson, W. K., "Linear algebra withapplications", 2nd ed., Chapters 1-7.Evaluation Method:* WeBWorKassignments 8%. There will be 4 Assignments posted every Thursdaywith a deadline of Tuesday. Your WeBWorK assignments are on * Writtenassginments 7%. There will be 2 such Assignments to be given onMay17 and May 25 and handed in by Friday May 22 and May 28, respectively.Solutionsto Midterm. MidtermGrades (out of 100)* Midtermexam (compulsory) 15%. To be given on Monday, May 17, 14:00-16:00.Room ARTS W-120. Please verify this information and bring yourMcGillID to the exam. The midterm will consist of one proof we did in class,one easy proof not done in class and exercises as in the assignments.* FinalExam 70%. To be given on Friday, May 28, 9:00-12:00, Room: LEA 14 & 15.Please verify this information and bring your McGill ID to the exam.Thestructure of the final will be similar to the midterm's; there will besome easy proofs, but most of the exam will consist of exercises verysimilarto those in the assignments and examples done in class. FinalExam CopyOffice Hours: Tue, Th 11:00-12:00.(Burnside Hall, 1108; 514-398-3815).Couse evaluations: Please fillthe course evaluations on line by loging in to your Minerva account.Thedeadline is Friday, May 28.Help Desk: There is a "helpdesk" which is a study center where advanced undergrads andgraduatestudents are at hand to assist you. This is highly recommended.
Linear Algebra And Its Applications 4th Edition Solutions.rarl
Detailed Syllabus( numbers refer toPoole's book) Date Material Comments WrittenAssignments Handouts May 3-7 Vectors:geometric and algebraic;adding and substracting vectors. Dot product; lengthsandangles. Lines and planes.Crossproduct. Linear equations.Methodsfor solving linear equations. Spanning sets andlineardependence. 1.1-1.3, 2.1-2.3 Students notfamiliar withcomplex numbers should read Appendix C in Poole. Knowledge of complexnumbersis assumed from the fourth lecture. Solvinglinear equations NewVersion May 12 May 10-14 Algebra withmatrices: addition,multiplication by scalar, multiplication, transpose. Elementarymatrics.The inverse matrix and its calculation by row-reduction; application tolinear equations. Subspace; row, column and null space of a matrix;basisand dimension. Linear transformations. Linear transformation andmatrices.Rotations. Reflections. Composition. 3.1-3.5 Algebrawith matrices May 17-21 Projections.Inverse lineartransformation. Eigenvalues and eigenvectors. Determinants. Laplace'sexpansions.Determinants of elementary matrices. The product (and other) formulafordeterminants. Calculation of determinants by row reduction. Determinantas a volume function. Cramer's rule and the adjoint matrix. Thecharacteristicpolynomial. Eigenvalues and eigenvectors revisited. Linear independenceof eigenspaces. Similarity and diagonalization. Representing lineartransformationsin different bases and diagonalization. Applications. 4.1-4.4 WrittenAss 1 May 24-28 Orthogonality inRn.Orthonormal bases. Orthonormal matrices and distance preservingtransformations.Orthogonal complements and orthogonal projections. The Gram-Schmidtprocess.A Symmetric matrice has real e.values and its e.spaces are orthogonaltoeach other. Orthogonal diagonalization. Applications: Quadratic formsandextrema of functions of 2 variables. 5.1-5.5 May 24 is VictoriaDay;make-up class is given ub 1B23 on May 25 and 26, 11:30 - 12:25. WrittenAss 2 Diagonalizationalgoirthms Notesfor Wednesday Lectures
Instructor: Dr. Eyal Z. Goren.Time: MWF 8:35-9:25 Location: BURN 920.Syllabus (in the large):We shall cover part of Chapter 10 andmost of chapters 12, 13, 14 of Dummit and Foote.January: Introduction to modules.Modules over PID. Applications to linear transformations and finitelygeneratedabelian groups. Time permitting: Smith's normal form.February-April: Introduction tofiled theory. Algebraic and transcendental extensions; separable andinseparableextensions. Splitting fields and algebraic closure. Galois groups. Thefundamental theorem of Galois theory. Applications to solving equationsby radicals. Finite and cyclotomic fields. Time permitting:InfiniteGalois groups. The inverse Galois problem.Prerequisite:MATH 251, MATH 370 (or equivalent courseswith my permission. Students that haven't taken a course on vectorspacescan still enrol, given my permission, but are advised to catch up onthismaterial by reading Dummit and Foote Sections 11.1 - 11.4).Note: This course is normally takenby honours students, though I do not consider that a requisite. One cantherefore expect it to be exciting, inspiring but also challenging.Text Book:* Dummitand Foote/ Abstract Algebra(Third Edition), Wiley.Other texts (on reserve at Schulich):* M. Artin / Algebra.* S. Lang / Algebra.* N. Jacobson / Abstract algebra* I.Stewart / Galois TheoryEvaluation Method:* 20%Assignments (12 weekly, short assignments. Handed-out and submitted on Mondays.Submitall. You may work together on your assignment, but in the end each hasto write his or her own solutions; identical assignments will be markedas zero.)*25% Midterm (Tuesday,March9, 17:30 - 19:00 BURN 1120). The topics are General Theory ofModules,Modules over a PID and General Theory of Fields. The material includesthose parts of Chapters 10 and 12 covered in class, and sections 13.1 -13.4, 13.6 in Chapter 13 of Dummit and Foote. See detailed syllabusbelow.* 55% FinalExam (Tuesday, April 20, 14:00, MAASS 328) -- If final exam grade is betterthan midterm then midterm doesn't count. Assignment grades always count(even in deferred/supplamental).Office Hours: Wednesday,Friday 9:30-11:00. (Burnside Hall, 1108; 514-398-3815).
Detailed Syllabus Date Material Comments Assignmentsand Solutions 9/3 Introduction.Groups. Subgroups.Order of an element and the subgroup is generates. Subroup generated bya set. The groups Z, Z/nZ, Z/nZ*. The Dihedral group D2n. 9/8 The Symmetricgroup Sn (cycles,sign, transpositions, generators). The group GLn(F). Thequaterniongroup Q. Groups of small order. Direct products. The subgroups of(Z/2Z)2. Cyclic groups and the structure of their subgroups.The group F* is cyclic. Commutator, centralizer and normalizersubgroups.Cosets. Refresh yourmemory of thesymmetric group. Assignment1 Solutions 9/15 Cosets.Lagrange'sTheorem. Normal subgroups and Quotient groups. Abelianization.Homomorphism,kernels and normal subgroups. The first homomorphism theorem. Inquestion 3) (2),p is a prime. Assignment2 Solutions 9/22 Thehomomorphism theorems(cont'd). The lattice of subgroups. Group actions on sets: actions,stabilizersand orbits. Examples. Assignment3 Solutions 9/29 Group actions onsets (cont'd):Cayley's theorem. The Cauchy-Frobenius formula. Applications tocombinatorics:necklaces designs, 14-15 square, Rubik's cube. Conjugacy classes in Sn. Assignment4 Solutions 10/6 Conjugacyclasses in An.Thesimplicity of An. The class equation. p-groups. In question 1,the groupG acts linearly on the vector space V. Assignment5 Solutions You can hand inyourassignment 5 on Wednesday October 15. 10/13 Free groups andBurnside'sproblem. Cauchy's Theorem. Syllow's Theorems -- statement andexamples. Assignment6 Solutions 10/20 Syllow'sTheorems -- proofand applications (e.g., groups of order pq and p2q).Finitelygenerated abelian groups. This version--> of theassignment correct typos of the one given in class. Assignment7 Solutions Numberof Groups of order N 10/27 Semi-directproducts andgroups of order pq. Groups of order less than 16. Composition series.TheJordan Holder Theorem. Assignment8 Solutions 11/3 Solvablegroups. Rings- basics. Ideals and quotient rings. Examples: Z, Z/nZ, R[x], R[[x]],R((x)). Midterm onMonday, November3 17:05-18:25, ARTS 210 MidtermSolutions MidtermGrades 11/10 Examples: Mn(R),Quaternions. Creating new rings: quotient, adding a free variable,fieldof fractions. Ring homomorphisms. First isomorphism theorem. Behaviorofideals under homomorphisms. In question 2, part (1), assume R is an integral domain! Assignment9 Solutions 11/17 More on ideals:intersection,sum, product, generation, prime and maximal. The Chinese RemainderTheorem.Euclidean rings. Examples: Z, F[x], Z[i]. PID's. Euclidean implies PID.Greatest common divisor and the Euclidean algorithm. Assignment10 Solutions 11/24 The Euclideanalgorithm.Prime and irreducible elements + agree in PID. UFD's. Prime andirreducibleagree in UFD. PID implies UFD. g.c.d. in a UFD. Gauss's Lemma. Assignment11 Solutions 12/1 R UFDimplies R[x]UFD. Existence of splitting fields. Construction of finite fields.
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