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MTH102

Linear Algebra, Ordinary Differential Equations and Laplace Transform

Module 1: Matrices and System of Linear Equations

Matrices, System of linear equations, Gauss elimination method, Elementary matrices. 1

Elementary matrices, Invertible matrices Gauss-Jordon method for finding inverse ofa matrix.

1

Determinants, Basic properties of determinants. 2

Cofactor expansion, Determinant method for finding inverse of a matrix, Cramer’sRule.

1

Rank of a matrix, Solvability of a system of linear equations, some applications 1

Module 2: Finite Dimensional Vector Spaces and Linear Transformations

Vector space, Subspace, Examples. 2

Linear span, Linear independence and dependence, Examples. 1

Basis, Dimension, Extension of a basis of a subspace, Intersection and sum of twosubspace, Examples, Row and column spaces, Application to the understanding of Rn.

3

Linear transformation, Kernel and Range of a linear map, Rank-Nullity Theorem. 2

Module 3: Inner Product Spaces, Eigenvalues, Eigenvectors and Diagonalizability

Inner product on Rn , Cauchy-Schwartz inequality, Orthogonal basis, Gram-Schmidtorthogonalization process.

2

Orthogonal projection, Orthogonal complement, Projection Theorem, FundamentalSubspaces.

1

Fundamental subspaces and their relations, An application (Least square solutions andleast square fittings).

1

Eigenvalues, Eigenvectors, Characterization of a diagonalizable matrix. 2

Diagonalization: Example, An application. 1

Diagonalization of a real symmetric matrix, Eigenvalues of Hermitian and UnitaryMatrices.

1

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Module 4: Introduction to Ordinary Differential Equations

Introduction to DE, Order of DE, First Order ODE Fx,y,y'=0. 1

Concept of solution (general solution, singular solution, implicit solution etc.),Geometrical interpretations (direction fields, nullclines)

1

Separable form, Reduction to separable form, Exact equations, Integrating factors[of the form F(x) and F(y)].

1

Linear equations, Bernoulli equation, orthogonal trajectories. 1

Picard’s existence and uniqueness theorem (without proof), Picard’s iterationmethod.

1

Numerical methods: Euler’s method, improved Euler’s method. 1

Module 5: Introduction to Second Order Ordinary Differential Equations

Second order linear ODE: fundamental system and general solutions ofhomogeneous equations, Wronskian, reduction of order. 1

Characteristic equations: real distinct roots, complex roots, repeated roots. 1

Non-homogeneous equations: undetermined coefficients 1

Non-homogeneous equations: variation of parameters 1

Extension to higher order differential equations, Euler-Cauchy equation. 1

Module 6: Power Series Solution, Special Functions and Strum-Liouville Boundary

Value Problems

Power series solutions: ordinary points (Legendre equation). 1

Power series solutions: regular singular points (Bessel equation), Frobeniusmethod, indicial equations. 1

Legendre polynomials and properties 1

Bessel functions and properties 1

Sturm comparison theorem, Sturm-Liouville boundary value problems, orthogonalfunctions.

2

Module 7: Laplace Transform

Laplace transform: Laplace and inverse Laplace transforms, first shiftingtheorem, existence, transforms of derivative and integral

1

Laplace transform: Differentiation and integration of transforms, unit step

function, second shifting theorem. 1Laplace transform: Convolution and applications, initial value problems 1

TOTAL 42