Linear Algebra and Matrix Theory

Syllabus Details

• Unit I:(Systems of Equations and Vector Spaces) System of linear equations, solutions of systems of linear equations, homogeneous system; matrices: elementary operations, reduced row-echelon form; concept of fields with examples; vector spaces and subspaces with examples; direct sum and algebra of subspaces, viz. sum, intersection, union, etc.; linear combinations, spanning sets, linear spans; linear dependence and independence in vector spaces; row and column space of a matrix; basis and dimension.
• Unit II:(Linear Transformations and Inner Product Spaces) Linear transformations; kernel and image of a linear transformation; rank and nullity; matrix representation of a linear operator; change of basis; similarity; dual space; inner product spaces with examples; Cauchy–Schwarz inequality with applications; orthogonality; orthonormal sets and bases; Gram–Schmidt orthogonalization process; adjoint operator.
• Unit III:(Eigenvalues and Matrix Decompositions) Eigenvalues and eigenvectors; Cayley–Hamilton theorem; algebraic and geometric multiplicity of characteristic roots; diagonalization of matrices; determinants; relating trace and determinant with eigenvalues; spectral decomposition of a symmetric matrix (full rank and non-full rank cases), example of spectral decomposition; spectral decomposition of an asymmetric matrix; factorization of a matrix; eigenvalues and eigenvectors for solution of differential equations.
• Unit IV:(Generalized Inverses and Quadratic Forms) Generalized inverse of a matrix; different classes of generalized inverse; properties of g-inverse; reflexive g-inverse, left weak and right weak g-inverse; Moore–Penrose (MP) g-inverse and its properties; real quadratic forms; linear transformation of quadratic forms; index and signature; reduction of quadratic form into sum of squares; Gram matrix with examples; Jordan canonical form.

Course Learning Objective

1. Visualize real vector spaces of higher but finite order, and distances between vectors and angles between them via inner product.
2. Explain linear independence, dimension of a vector space, and linear spans of collections of vectors, and display subspaces.
3. Apply the concepts of linear transformations on vectors, and determine whether exact solutions to systems of linear equations with multiple dimensions/variables exist, and find them when they do.
4. Work with real matrices: understand column spans and row spans of matrices, and prove results on eigenvalues and quadratic forms.
5. Explain what generalized inverses of matrices are, and prove results and solve problems where these are needed.

Recommended Books

1. Biswas, S. A Text Book of Matrix Algebra. 2nd ed. New Age International Publishers, 1997
2. Golub, G. H. and Van Loan, C. F. Matrix Computations. 2nd ed. Johns Hopkins University Press, Baltimore-London, 1989
3. Hadley, G. Linear Algebra. Narosa Publishing House (Reprint), 2002
4. Hoffman, K. and Kunze, R. Linear Algebra. 2nd ed. Prentice-Hall, New Delhi, 1978
5. Robinson, D. J. S. A Course in Linear Algebra with Applications. World Scientific, Singapore, 1991
6. Rao, C. R. Linear Statistical Inference and Its Applications. 2nd ed. John Wiley & Sons, 1973
7. Searle, S. R. Matrix Algebra Useful for Statistics. John Wiley & Sons, 1982
8. Strang, G. Linear Algebra and Its Applications. 2nd ed. Academic Press, London New York, 1980