Course Archives Theoretical Statistics and Mathematics Unit | ||||||||
Course: Linear Algebra II Instructor: B V Rajarama Bhat Room: G24 Level: Undergraduate Time: Currently offered |
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Syllabus Past Exams Syllabus: Determinant of n-th order and its elementary properties, expansion by a row or column, statement of Laplace expansion, determinant of a product, statement of Cauchy- Binet theorem, inverse through classical adjoint, Cramers rule, determinant of a partitioned matrix, Idempotent matrices. Norm and inner product on Rn and Cn, norm induced by an inner product, Orthonormal basis, Gram-Schmidt orthogonalization starting from any finite set of vectors, orthogonal complement, orthogonal projection into a subspace, orthogonal projector into the column space of A, orthogonal and unitary matrices. Characteristic roots, relation between characteristic polynomials of AB and BA when AB is square, Cayley-Hamilton theorem, idea of minimal polynomial, eigenvectors, algebraic and geometric multiplicities, characterization of diagonalizable matrices, spectral representation of Hermitian and real symmetric matrices, singular value decomposition. Quadratic form, category of a quadratic form, use in classification of conics, Lagranges reduction to diagonal form, rank and signature, Sylvesters law, determinant criteria for n.n.d. and p.d. quadratic forms, Hadamards inequality, extrema of a p. d. quadratic form, simultaneous diagonalization of two quadratic forms one of which is p.d., simultaneous orthogonal diagonalization of commuting real symmetric matrices, square-root method. Note: Geometric meaning of various concepts like subspace and flat, linear independence, projection, determinant (as volume), inner product, norm, orthogonality, orthogonal projection, and eigenvector should be discussed. Only finite-dimensional vector spaces to be covered. 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. Evaluation:
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