MA2014E MATHEMATICS IV
| L |
T |
P |
O |
C
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| 3 |
1* |
0 |
5 |
3
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| Total Lecture Sessions: 39
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Course Outcomes
| Outcome
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Description
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| CO1
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Characterise a linear system in terms of vectors, linear combinations, and span.
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| CO2
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Understand the concepts of projection, orthogonality, and linear transformations of vectors.
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| CO3
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Understand convergence, continuity, and differentiability of real valued functions.
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| CO4
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Construct optimization problems for real world applications.
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| CO5
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Understand the existence and uniqueness of the solutions for formulated optimization problems.
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Syllabus
Linear Algebra
- Vector spaces
- Subspaces
- Linear Independence
- Basis
- Dimension
- Inner product spaces
- Norms
- Orthogonality
- Gram-Schmidt Orthogonalization
- Projection
- Least squares approximations
- Linear transformations
- Kernel and Image
- Rank-nullity theorem
- Matrix representation of linear transformation
- Change of Basis
- Type of Matrices
- Singular Value Decomposition
Real Analysis
- Real Line
- Open and Closed Sets
- Sequences and Sub-sequences
- Compact and Connected Sets
- Continuous Mapping
- Boundedness on Compact Sets
- Convergence
- Differentiable Mapping
- Chain Rule
- Mean-Value Theorem
- Taylor’s Theorem and Higher Derivatives
Fundamentals of Optimization
- Convex sets and Convex cones
- Affine sets
- Convex and Concave functions
- Global vs Local optimality
- Quadratic Functions
- Linear Programming: Introduction, Optimization model, Formulation, Geometric Ideas and applications.
- Nonlinear Programming
- Unconstrained Optimization: Conditions of maxima and minima for unconstrained optimization.
- Numerical methods for unconstrained optimization: Line search methods; Method of Steepest Descent and Newton’s method.
- Constrained Optimization: Conditions of maxima and minima for constrained optimization, Method of Lagrange Multipliers, Karush-Kuhn-Tucker theory.
References
- Jerrold E. Marsden and Michael J. Hoffman. Elementary Classical Analysis. Macmillan, 1993.
- R. K. Jain and S. R. K. Iyengar, Advanced Engineering Mathematics, 5th Edn, Narosa Publication, 2016.
- E. Kreyszig, H. Kreyszig, E. J. Norminton, Advanced Engineering Mathematics, 10th Edn, Wiley, New Delhi, 2015.
- Edwin K. P. Chong and Stanislaw K. Zak, An Introduction to Optimization, 2nd Edn, Wiley, 2005.
- David G. Luenberger and Yinyu Ye. Linear and Nonlinear Programming. Vol. 2. Reading, MA: Addison-Wesley, 1984.
- G Mohan and Kusum Deep, Optimization Techniques, New age International Publishers, 2009
