Course Archives: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore

Course Archives Theoretical Statistics and Mathematics Unit

Course: Linear Algebra I
Level: Undergraduate
Time: Currently not offered

Syllabus:

Homogeneous and non-homogeneous systems of linear equations, condition for consistency, solution set as a translate of a subspace.
Vector spaces, subspaces, linear independence, span, basis and dimension, sum and intersection of subspaces, direct sum, complement and projection.
Linear transformation and its matrix with respect to a pair of bases, properties of matrix operations, use of partitioned matrices.
Column space and row space, rank of a matrix, nullity, rank of AA*.
g-inverse and its elementary properties, left inverse, right inverse and inverse, inverse of a partitioned matrix, lower and upper bounds for rank of a product, rankfactorization of a matrix, rank of a sum. Elementary operations and elementary matrices, Echelon form, Normal form, Hermite canonical form and their use in solving linear equations and in finding inverse or g-inverse. LDU-decomposition.

Note: The field of scalars should be assumed to be subfields of complex numbers, i.e., subsets closed under addition, subtraction, multiplication and division by a nonzero number. The main examples to be considered should be the field of real, complex or rational numbers.

Reference Texts:

(a) C. R. Rao: Linear Statistical Inference and its Applications.
(b) A. Ramachandra Rao and P. Bhimasankaram: Linear Algebra.
(c) K. Hoffman and R. Kunze: Linear Algebra.
(d) F. E. Hohn: Elementary Matrix Algebra.
(e) P. R. Halmos: Finite Dimensional Vector Spaces.
(f) S. Axler: Linear Algebra Done Right!
(g) H. Helson: Linear Algebra.
(h) R Bapat: Linear Algebra and Linear Models.
(i) R. A. Horn and C. R. Johnson: Matrix Analysis.
(j) M. Artin: Algebra.

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