# Linear Algebra, Ordinary Differential Equations and Laplace Transform Module

May 29th, 2015
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Module 1: Matrices and System of Linear Equations Matrices, System of linear equations, Gauss elimination method, Elementary matrices. Elementary matrices, Invertible matrices Gauss-Jordon method for finding inverse of a matrix.

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MTH102 Linear Algebra, Ordinary Differential Equations and Laplace TransformTopicNumber of LecturesModule 1: Matrices and System of Linear EquationsMatrices, System of linear equations, Gauss elimination method, Elementary matrices.1Elementary matrices, Invertible matrices Gauss-Jordon method for finding inverse of a matrix.1Determinants, Basic properties of determinants.2Cofactor expansion, Determinant method for finding inverse of a matrix, Cramer's Rule.1Rank of a matrix, Solvability of a system of linear equations, some applications1Module 2: Finite Dimensional Vector Spaces and Linear TransformationsVector space, Subspace, Examples.2Linear span, Linear independence and dependence, Examples.1Basis, Dimension, Extension of a basis of a subspace, Intersection and sum of two subspace, Examples, Row and column spaces, Application to the understanding of Rn.3Linear transformation, Kernel and Range of a linear map, Rank-Nullity Theorem.2Module 3: Inner Product Spaces, Eigenvalues, Eigenvectors and DiagonalizabilityInner product on Rn , Cauchy-Schwartz inequality, Orthogonal basis, Gram-Schmidt orthogonalization process.2Orthogonal projection, Orthogonal complement, Projection Theorem, Fundamental Subspaces.1Fundamental subspaces and their relations, An application (Least square solutions and least square fittings).1Eigenvalues, Eigenvectors, Characterization of a diagonalizable matrix.2Diagonalization: Example, An application.1Diagonal

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