Course

Syllabus

Systems of Linear Equations and Matrices: Vectors; Matrices; Systems of Linear Equations; Echelon form; Row reduction; Gaussian elimination method.

Vector Spaces: Vector spaces; Sub spaces; Linear dependence and independence; Basis and Dimension; Fundamental Matrix Spaces; Rank-Nullity Theorem.

Inner Product Spaces: Inner products; Orthogonality; Orthogonal basis; Orthogonal complements; Projection on subspace; Gram Schmidt Process; QR Decomposition; Least Square Principle; Least Square Fitting to Data.

Linear Transformations: Linear transformation; Relation between matrices and linear transformations; Kernel and range of a linear transformation; Rank-Nullity Theorem, Change of basis; Orthogonal transformations and rotations; Nilpotent transformations, Unitary, Hermitian, and skew Hermitian matrices or operators; Self-Adjoint and Normal operators.

Eigen values and Eigen vectors: Eigenvalues and Eigenvectors; Diagonalization; Orthogonal Diagonalization; Quadratic Forms; Similarity of linear transformations; Diagonalisation and its applications – Jordan canonical form. LU-Decomposition, Singular Value Decomposition.

Lab Experiments:

1. Matrix operations,Finding determinant, rank, inverse, Generation of random matrices with given rank
2. Solving systems of linear equations using Gaussian elimination
3. Transforming a set of vectors into an orthonormal basis.
4. Factoring a matrix into lower and upper triangular matrices.
5. Factoring a matrix into an orthogonal matrix and an upper triangular matrix.
6. Compute eigenvalues/eigenvectors of matrices and visualize their geometric meaning.
7. Compute large matrix powers by diagonalizing a matrix
8. Computing SVD for dimension reduction and data analysis.
9. Least Square Fitting to Data.
10. Scaling, Shifting, Rotation of imagesusing Linear Transformations

Text Books / References

Text Book:

• Howard Anton and Chris Rorres, “Elementary Linear Algebra”, Tenth Edition, John Wiley & Sons, 2010.

References:

1. Nabil Nassif, Jocelyne Erhel, Bernard Philippe, Introduction to Computational Linear Algebra, CRC press, 2015
2. Sheldon Axler, Linear Algebra Done Right, Springer, 2014. Gilbert Strang, “Linear Algebra for Learning Data”, Cambridge press, 2019.

Course Outcomes:

• CO1: Solve systems of linear equations using matrix operations and row reduction methods.
• CO2 : Understand vector spaces, subspaces, basis, dimension, and rank–nullity concepts.
• CO3: Apply inner product and orthogonality concepts including projections and least squares methods.
• CO4 : Understand and apply concepts of vector-valued functions, space curves, curvature and motion in space. Apply partial derivatives for tangent planes and Hessian matrix.
• CO5 : Compute eigenvalues and eigenvectors and perform diagonalization and matrix decompositions.

CO-PO Mapping: