Applied Mathematics Course Contents

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1 Applied Mathematics Course Contents Matrices, linear systems Gauss-Jordan elimination, Determinants, Cramer's rule, inverse matrix. Vector spaces, inner product and norm, linear transformations. Matrix eigenvalue problem, orthogonal matrices, diagonalization. Function spaces, ortogonal and orthonormal set of functions. Second order linear ordinary differential equations, Initial and boundary value problems, solution by variation of parameters. The Sturm-Liouville problem, eigenvalues and eigenfunctions, orthogonal eigenfunctions expansion. Partial differential equations, initial and boundary conditions, vibrating string, wave equation The method of sepation of variables, use of Fourier series. Solution of Initial and boundary value problems for (heat) diffusion equation, wave equation, Laplace equation in a bounded domain. Fourier integrals, heat equations in the whole and half space, wave equation in unbounded domains, use of Fourier integrals.

2 Matrices and Linear Systems Roughly speaking, matrix is a rectangle array. We shall discuss existence and uniqueness of solution for a system of linear equation. The method of Gauss ellimination will be given to solve the system.

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29 18 6x 4x 6 3 2x 3 4x x 0 3x x 1 3 2x 3.Show that theyare inconsistent systems , echelon forms of Find row reduced x x

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32 Vector Spaces A quantity such as work, area or energy which is described in terms of magnitude alone is called a scalar. A quantity which has both magnitude and direction for its describtion is called a vector. A vector is an element of vector space.

33 Definiton: A vector space V in R is the set satisfying If u, v V and a R, then u v, au V. u v v u u ( v w) ( u v) w u 0 u (a unique zero element) u ( u) 0 (a uique additive inverse) ( a b) u au bu ( ab) u a( bu) 1u u (here1 is identity scalar) a( u v) au av

34 Examples for vector spaces V R R P F[a, b] (function - space on interval[a, b]) C n {0} n mxn n (space of matrices) [x] (space of polynomials) [a, b] (continously differentiable func.)

35 Linear Dependence Define a c where If has u 1 1 if u 1 the equation, u only trivial 2 any c 2 u,... u c i linear combination of u i 2 V, c n is... c i c 1 u n 1 u n, R, i c 2 u 1,2,3, c solution for all nonzero vectors then are linealy independent.otherwise, zero then they are linearly dependent. c n i u, n 0

36 Page 297 (2) Example: A= Rank of A is two because the first two rows are linearly independent.

37 to equivalent row is : ranka A Example

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39 Dimension of a vector space V SpanS= All linear combinations of vectors of the subset S of V. A basis for V is a linearly independent subset S of V which spans the space V. That is, SpanS= V where S is linearly independent. dimv= The number of vectors in any basis for V. V is finite-dimensional if V has a basis consisting of a finite number of vectors.

40 Note: (6) is known as the dimension theorem

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47 ., , where, are All solutions with : R c c x x x x x x x x x x x x x x x x x x Example T h T h

48 Determinant Determinant can be defined by a function form square matrices to scalars. Our efficient computational procedure will be cofactor expansion.

49 Page 306a Define det[ a 11 ] a 11. A determinant of second order is denoted and defined by D det A a a a a a 11 a 22 a 12 a 21

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60 Inverse of a Matrix. Gauss-Jordan Ellimination. 318a Find inverse of A if its inverse exists A Dosome elementary row operations as [ A, I] [ I, A 1 ].

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70 Examples

71 Linear Transformations transform. integral transform, differential rotation, projection, reflection, multiple operator, - scalar operator, identity Zero transform, : Examples, scalar a and, in and vectors all for if linear transformation a called is cf(u). F(cu) F(v) F(u) v) F(u c V v u F

75 Range and Null (Kernel) spaces V. nullityf rankf Theorem rankf (RangeF) nullityf (NullF) W V u u F v v RangeF V u u F u NullF linear transform. W V F dim : dim dim vectors. images all includes } ), ( : { }. 0, ) ( : { be a : Let

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Applied Mathematics Course Contents Matrices, linear systems Gauss-Jordan elimination, Determinants, Cramer's rule, inverse matrix.

Applied Mathematics Course Contents Matrices, linear systems Gauss-Jordan elimination, Determinants, Cramer's rule, inverse matri. Vector spaces, inner product and norm, linear transformations. Matri eigenvalue