Linear Algebra

Review of matrices and linear systems of equations.

Vector Spaces

Vector spaces – Sub spaces – Linear independence – Basis – Dimension – Inner products – Orthogonality – Orthogonal basis – Gram Schmidt Process – Change of basis. Orthogonal complements – Projection on subspace – Least Square Principle.

Linear Transformations

Positive definite matrices – Matrix norm and condition number – QR- Decomposition – Linear transformation – Relation between matrices and linear transformations – Kernel and range of a linear transformation.

Optimization

Introduction: Optimization – optimal problem formulation, engineering optimization problems, optimization algorithms, numerical search for optimal solution.

Single Variable optimization: Optimality criteria, bracketing methods – exhaustive search method, bounding phase method- region elimination methods – interval halving, Fibonacci search, golden section search, point estimation method- successive quadratic search, gradient based methods.

Multivariable Optimization: Optimality criteria, unconstrained optimization – solution by direct substitution, unidirectional search – direct search methods evolutionary search method, simplex search method, Hook-Jeeves pattern search method, gradient based methods – steepest descent, Cauchy’s steepest descent method, Newton’s method, conjugate gradient method – constrained optimization. Kuhn-Tucker conditions.

Course Objectives

The course is expected to enable the students

• To solve simultaneous algebraic equations using methods of matrix algebra.
• To use vector space methods and diagonalization in practical problems.
• To understand the concept of search space and optimality for solutions of engineering problems.
• To understand some computation techniques for optimizing single variable functions.
• To carry out various computational techniques for optimizing severable variable functions.

Course Outcomes

• CO1: Understand the basic concepts of vector space, subspace, basis and dimension.
• CO2: Understand the basic concepts of inner product space, norm, angle, Orthogonality and projection and implementing the Gram-Schmidt process, to obtain least square solution.
• CO3: Understand the concept of linear transformations, the relation between matrices and linear transformations, kernel, range and apply it to change the basis, to get the QR decomposition, and to transform the given matrix to diagonal/Jordan canonical form.
• CO4: Understand different types of Optimization Techniques in engineering problems. Learn Optimization\ methods such as Bracketing methods, Region elimination methods, Pointestimation methods.
• CO5: Understand the Optimality criteria for functions in several variables and learn to apply OT methods like unidirectional search and direct search methods.
• CO6: Learn constrained optimization techniques. Learn to verify Kuhn-Tucker conditions and Lagrangian Method.

CO – PO Mapping

Textbook(s)

• Howard Anton and Chris Rorrs, “Elementary Linear Algebra”, Ninth Edition, John Wiley & Sons, 2000.
• S.S. Rao, “Optimization Theory and Applications”, Second Edition, New Age International (P) Limited Publishers, 1995.

Reference(s)

• D. Poole, Linear Algebra: A Modern Introduction, 2nd Edition, Brooks/Cole, 2005.
• Gilbert Strang, “Linear Algebraand its Applications”, Third Edition, Harcourt College Publishers, 1988.
• Kalyanmoy Deb, “Optimization for Engineering Design Algorithms and Examples”, Prentice Hall of India, New Delhi, 2004.
• Edwin K.P. Chong and Stanislaw H. Zak, “An Introduction to Optimization”, Second Edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, 2004.
• M. Asghar Bhatti, “Practical Optimization Methods: with Mathematics Applications”, Springer Verlag Publishers, 2000.

Evaluation Pattern 50:50 (Internal: External)