Syllabus
Unit I
Vector Spaces: System of linear equations – consistency and solutions by Guess methods, Rank, row echelon form, reduced row echelon form.
(Sections: 1.1-1.7, 2.1)
Vector spaces, Sub spaces – Linear independence – Basis – Dimension, Null space, row space and column space (Sections: 4.1 – 4.8)
Inner Product Spaces: Inner products – Orthogonality – Orthogonal basis – Gram Schmidt Process – Change of basis – Orthogonal complements – Projection on subspace – Least Square Principle. QR- Decomposition. (Sections 6.1 – 6.4)
Unit II
Linear Transformations: Linear transformation – Relation between matrices and linear transformations – Kernel and range of a linear transformation – Change of basis – Nilpotent transformations. Symmetric and Skew Symmetric Matrices. (Sections 8.1 – 8.5, 5.1-5.2)
Unit III
Eigen values and Eigen vectors: Eigen Values and Eigen Vectors, Diagonalization, Orthogonal Diagonalization, Quadratic Forms, Diagonalizing Quadratic Forms, Canonical form , Similarity of linear transformations – Diagonalisation and its applications. (Sections 7.1 – 7.3)
Case Studies: Applications on least square and image transformations.
Lab Practice Problems: Matrices, Matrix operations. Solving system of linear equations, rank and nullity. Orthogonality. Matrix of linear transformations. Affine transformations, scaling, shifting and rotation of images. Eigen values and eigen vectors and matrix decompositions.
Text Books / References
TEXT BOOK:
1) Howard Anton and Chris Rorrs, “Elementary Linear Algebra”, Ninth Edition, John Wiley & Sons, 2000.
REFERENCE BOOKS:
1) D. Poole, Linear Algebra: A Modern Introduction, 2nd Edition, Brooks/Cole, 2005.
2) Gilbert Strang, “Linear Algebraand its Applications”, Third Edition, Harcourt College Publishers, 1988.
