NITH SYLLABUS

MA-221LINEAR ALGEBRA & APPLICATIONS
CreditsLTP
4310

UNIT-1

SYSTEM OF LINEAR EQUATIONS
  • Matrices and elementary row operations
  • Row reduced echelon matrices
  • Homogeneous system of linear equations
  • Elementary matrices
  • LU Decomposition
  • Some applications giving rise to Linear Systems Problems

UNIT-2

VECTOR SPACE
  • Fields
  • Fields of numbers
  • Finite fields
  • Vector Spaces over R and C
  • Subspaces
  • Linear independence
  • Basis and dimension of a vector space
  • Ordered basis and co-ordinates
  • Application to difference equations and markov chains

UNIT-3

LINEAR TRANSFORMATION
  • Linear transformations
  • Rank and Nullity of linear transformation
  • Algebra of linear transformation
  • Isomorphism
  • Invertible linear transformations
  • Dual and double dual of a vector space and transpose of a linear transformation
  • Matrix representation of Linear Transformation
  • Eigenvalue and eigenvector of an linear transformation
  • Discrete dynamical systems
  • Application to differential equations

UNIT-4

EIGEN DECOMPOSITION
  • Diagonalizability of linear operators of finite dimensional vector spaces
  • Simultaneous triangulization and simultaneous diagonalization
  • Primary decomposition theorem - diagonal and nilpotent parts
  • Applications to Image processing and Statistics

UNIT-5

INNER PRODUCT SPACES
  • Gram-Schmidt orthogonalization
  • Best approximation of a vector by a vector belonging a given subspace and application to least square problems
  • Adjoint of an operator
  • Hermitian
  • Unitary and normal operators
  • Singular Value Decomposition and its applications
  • Spectral decomposition
  • Applications of Inner product spaces
BOOKS
  1. Linear algebra and it’s applications by G. Strang, Cengage.
  2. Linear Algebra and its Applications by D.C. Lay, Pearson.
  3. Linear Algebra by K. Hoffman and R. Kunze, Pearson.
  4. Linear Algebra: A Modern Introduction by D. Poole, Brooks/Cole.
  5. Linear Algebra: A Geometric Approach by S. Kumaresan, Prentice-Hall of India.
  6. Linear algebra (undergraduate text in mathematics) by S. Lang, Springer
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