Linear Algebra

Syllabus Details

• Unit I: Matrices and elementary row/column operations, Row-reduced Echelon form, System of linear equations (homogeneous and non-homogeneous), Inverse of a matrix, Crammer rule
• Unit II: Vector Spaces and Subspaces, Basis and dimension, Linear Transformations and their matrix representations, Rank-Nullity Theorem, Quotient Spaces, Direct Sum, Dual of a Vector Space and Construction of Dual Basis.
• Unit III: Orthogonality, Cauchy-Schwarz inequality, Gram-Schmidt orthonormalization process
• Unit IV: Eigenvalues and eigenvectors, Minimal and Characteristic polynomials, Cayley-Hamilton Theorem, Diagonalizable and Triangulable operators, Companion matrix, Primary Decomposition Theorem and Cyclic Decomposition Theorem, The Rational and Jordan Canonical forms
• Unit IV: Matrix representation of bilinear forms, Orthogonality, classification of bilinear forms

Course Learning Objective

1. Solve systems of equations using basic matrix theory.
2. Explain Vector Spaces and Linear Transformations with matrix representations.
3. Find Eigenvalues and eigenspaces of a linear transformation
4. Find Diagonal/Triangular form of a Matrix.
5. Analyze a linear transformation by finding Rational and Jordan Canonical forms.
6. Apply Gram-Schmidt process of orthonormalizations in Inner Product Spaces and solve related problems.

Recommended Books

1. K. Hoffman and R. Kunze, Linear Algebra (2nd edition), Prentice-Hall of India, 2005
2. Vivek Sahai, Vikas Bist, Linear Algebra, Narosa Publishing House
3. M. Artin, Algebra, Prentice-Hall of India, 2005
4. A. R. Rao and P. Bhimashankaram, Linear Algebra, second edition, TRIM series, 2005