NPTEL Lecture on Linear Algebra

Course Introduction

• Fundamental course in both pure and applied mathematics.
• Essential for problems in engineering and various disciplines.
• Focus on proving theorems and understanding linear algebra concepts.
• Course divided into 13 modules.
• Recommended books:
  • Paul Halmos: Finite Dimensional Vector Spaces (Springer, 2011)
  • Hoffman and Kunze: Linear Algebra (Prentice Hall, 2004)

Module Overview

Module 1: Systems of Linear Equations

• Lectures 2-7
• Elementary row operations and equivalence of systems.
• Gaussian elimination process.
• Row reduced echelon matrices and elementary matrices.
• Homogeneous and non-homogeneous equations.
• Characterizing solutions using matrix invertibility.

Module 2: Vector Spaces

• Lectures 8-10
• Axiomatic definition and examples.
• Subspaces and spanning sets.
• Linear independence of vectors.

Module 3: Basis and Dimension

• Lectures 10-13
• Linear dependence and independence.
• Spanning subsets and basis.
• Determining the dimension of subspaces.

Module 4: Linear Transformations

• Lectures 14-18
• Definition and examples.
• Null space and range space.
• Rank Nullity Dimension Theorem.
• Row rank and column rank.

Module 5: Matrix of a Linear Transformation

• Lectures 18-20
• Matrix representation of a vector and linear transformations.
• Composition and inverse transformations.
• Similarity transformations.

Module 6: Linear Functionals and Dual Spaces

• Lectures 21-25
• Linear functional and representation theorem.
• Dual space and dual basis.
• Annihilators and double annihilators.

Module 7: Eigenvalues and Eigenvectors

• Lectures 26-29
• Matrix formulation and diagonalizability.
• Characteristic polynomial and eigenspaces.
• Minimal polynomial and Cayley Hamilton theorem.

Module 8: Invariant Subspaces and Triangulability

• Lectures 30-32
• Invariant subspaces and T-conductor.
• Triangulability vs diagonalizability.
• Projection operators.

Module 9: Direct Sum Decompositions

• Lectures 33-34
• Direct sum decompositions of vector spaces.
• Invariance and diagonalizability.

Module 10: Primary and Cyclic Decomposition Theorems

• Lectures 35-38
• Primary decomposition theorem.
• Jordan decomposition and cyclic decomposition.

Module 11: Inner Product Spaces

• Lectures 39-42
• Inner product and norm of vector spaces.
• Orthogonality, orthonormality.
• Gram-Schmidt process and QR decomposition.

Module 12: Best Approximation

• Lectures 43-45
• Best approximation in inner product spaces.
• Least square solutions and orthogonal projections.

Module 13: Adjoint of an Operator

• Lectures 46-47
• Adjoint operator properties.
• Inner product space isomorphisms.

Module 14: Self-Adjoint, Normal, and Unitary Operators

• Lectures 48 onwards
• Unitary operators and unitary equivalence.
• Self-adjoint operators and spectral theorem.
• Normal operators and spectral theorem.

Conclusion

• The course will cover all essential topics in a first course on linear algebra.
• Detailed study and examples provided throughout the 14 modules.
• Actual lectures will begin from the next session.