Advanced Linear Algebra

ABOUT THE COURSE: This course is designed to provide students with an understanding of Mathematical concept on Linear algebra that includes basic as well as advanced level. Attempt is taken to cover both Both theoretical as well as computation perspectives. There are six componenets: i) Linear System of equations, (ii) Vector spaces, (iii) Linear transformations, (iv) Cannonical forms and Jordan forms, (v) Inner product spaces and different operators in it, (vi ) Bilinear and Quadratic forms,Orthogonal projection and Spectral theory, and (vii) Singular value decomposition.INTENDED AUDIENCE: Master students of Mathematics,,Physics, B.Tech III Year Electrical, Computer Science,PREREQUISITES: Linear Algebra B.Sc /BE I/II year

1.1 System of linear equation (Review)1.2 Elementary matrix operation, elementary matrices,1.3 Rank of matrix, Matrix inverse1.4 Vector spaces,1.5 Subspaces1.6 Bases, and dimension1.7 Ordered basis, coordinate matrix1.8 Computation concerning subspaces1.9 Linear transformation,1.10 Existence of Linear transformation1.11 Rank Nullity theorem(Review)1.12 Representation of transformations by matrices1.13 Change of ordered basis and matrix representation of transformations1.14 Algebra of Linear Transformation1.15 Linear operators and Linear Functional1.16 Dual Space, Double dual spaces1.17 Transpose of Linear transformation1.18 Characteristic values, diagonalization, (Review)1.19 Annihilating polynomials1.20 Invariant subspace1.21 Triangulation (2)1.22 Simultaneous triangulation and simultaneous diagonalization1.23 Direct sum decomposition1.24 Invariant direct sums1.25 The primary decomposition theorem1.26 Jordan forms1.27 Rational form1.28 Inner product spaces1.29 Gramian matrix, Gram Schmidt orthogonalization1.30 Orthogonal complements, Best approximation1.31 Operators on Inner product spaces1.32 The adjoint of a linear operator1.33 Normal and self adjoint operator1.34 Unitary and orthogonal operators and their matrices1.35 Bilinear and Quadratic forms1.36 Orthogonal projections and the spectral theorem1.37 *Generalized g-inverse of a matrix, The Singular value decomposition