COURSE UNIT TITLE

: LINEAR ALGEBRA II

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 2038 LINEAR ALGEBRA II COMPULSORY 4 0 0 7

Offered By

Mathematics (English)

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics (English)
Mathematics (Evening)

Course Objective

The focus of this course will be on abstract vector spaces, linear operators, canonical forms, inner product spaces and bilinear forms. Students will be expected to learn the important theorems of linear algebra and understand their proofs.

Learning Outcomes of the Course Unit

1   be able to identify and use eigenvalues and related eigenvectors, and the characteristic and minimal polynomials of a linear operator.
2   be able to use diagonalizability, the Spectral Theorem for symmetric matrices over real numbers and its application to symmetric bilinear forms, quadratic forms, conic sections, quadric surfaces.
3   be able to use linear transformations and their matrix representations in bases interchangeably via the Change-of-Basis formula, and linear functionals, dual spaces.
4   be able to find complex eigenvalues and some canonical forms like the Jordan canonical form.
5   be able to use inner product spaces over real or complex numbers, Gram-Schmidt Orthogonalization Process to obtain orthogonal and orthonormal bases for subspaces, orthogonal complements and the orthogonal projection map onto a finite-dimensional subspace.
6   be able to use the adjoint of a linear operator, self-adjoint (=Hermitian), unitary, orthogonal and normal operators and matrices and the Spectral Theorem for normal operators in inner product spaces over real and complex numbers.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Review of determinants: its Properties; Cofactors and Cramers rule; Signed area in R2 and signed volume in R3.
2 Abstract vector spaces over the field R and inner product spaces over R: Extending the concepts in Rn to abstract vector spaces over R; Linear independence, basis and dimension; Function spaces.
3 Inner product spaces; Orthogonal basis; Orthonormal basis; Gram-Schmidt Orthogonalization Process; Orthogonal projection onto a finite-dimensional subspace of an inner product space; Rotation maps in R2 and R3; Reflections in R2 and R3.
4 Matrix Representation of Linear Transformations: Linear transformations on abstract vector spaces; Isomorphisms of vector spaces; Coordinates of a vector with respect to a basis; Change of basis matrix; Matrices representing linear transformations with respect to given bases; Change of basis formula; Similar matrices.
5 Eigenvalues and Eigenvectors: The characteristic polynomial of a linear transformation; Diagonalizability; Subspaces invariant under a linear operator; Some applications.
6 Spectral Theorem for symmetric matrices over real numbers; Orthogonal diagonalization of symmetric matrices.
7 Quadratic forms.
8 Conic sections; Quadric surfaces.
9 Inner Product Spaces over real numners and complex numbers: Review of inner products and norms for real and complex vector spaces; The Gram-Schmidt Orthogonalization Process; Orthogonal complements; The adjoint of a linear operator.
10 Normal and self-adjoint (=Hermitian) operators and matrices; Unitary and orthogonal operators and matrices; Orthogonal projections and the Spectral Theorem.
11 Symmetric bilinear forms, quadratic forms; Nonorthogonal diagonalization of quadratic forms and Slyvester s Law of Inertia; Positive definite and positive semidefinite operators and matrices, Slyvester s Criteria for positivity.
12 Complex eigenvalues and the Jordan Canonical Form.
13 Direct sums; Invariant subspaces; The Cayley-Hamilton Theorem; The Minimal polynomial.
14 Linear functionals; Dual spaces and the double dual space. Some of the following further topics if time permits. Rational Canonical Form; Computer graphics and geometry; Isometries (rigid motions) of R2 and R3; Perspective projection and projective equivalence of conics; Affine and projective geometry; Matrix exponentials and differential equations; Lie groups of matrices.

Recomended or Required Reading

Textbook(s): Linear Algebra: A Geometric Approach, 2nd Edition; T. Shifrin, M.R. Adams, W.H. Freeman and Company, New York, 2010.

Supplementary Book(s):
1-Introduction to Linear Algebra, 5th Edition; Gilbert Strang, Wellesley-Cambridge Press, 2016.
2-Linear Algebra, 2nd Edition; Serge Lang, ADDISON-WESLEY PUBLISHING COMPANY.
3-Linear algebra, 4th Edition; S.H.Friedberg, A.J.Insel, L.E.Spence, Pearson, 2014.

Materials: Instructor's lecture notes and presentations

Planned Learning Activities and Teaching Methods

Lecture notes, presentation, problem solving, discussion.

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE MIDTERM EXAM
2 FIN FINAL EXAM
3 FCG FINAL COURSE GRADE MTE * 0.50 + FIN * 0.50
4 RST RESIT
5 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.50 + RST * 0.50


Further Notes About Assessment Methods

1 Midterm Exam
Final Exam

Assessment Criteria

%50 (Midterm examination) +%50 (Final examination)

Language of Instruction

English

Course Policies and Rules

You can be successful in this course by studying from your textbooks and lecture notes on the topics to be covered every week, coming to class by solving the given problems, establishing the concepts by discussing the parts you do not understand with your questions, learning the methods, and actively participating in the course.

Contact Details for the Lecturer(s)

Engin Mermut
e-mail: engin.mermut@deu.edu.tr
Phone: (232) 301 85 82

Office Hours

To be announced later.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 14 4 56
Preparations before/after weekly lectures 14 4 56
Preparation for final exam 1 30 30
Preparation for midterm exam 1 20 20
Final 1 2 2
Midterm 1 2 2
TOTAL WORKLOAD (hours) 166

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.1544343
LO.2544353
LO.3534343
LO.443434
LO.554434
LO.654434

CONTACT INFORMATION
Dokuz Eylül Üniversitesi Cumhuriyet Bulvarı No: 144 35210 Alsancak / İZMİR
Phone: +90(232) 412 12 12 - Fax: +90 (232) 464 81 35

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