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

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1 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 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 epansion. 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, matri is a rectangle array. We shall discuss eistence 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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4 Pages Advanced Engineering Mathematics by Erwin Kreyszig

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23 ii) i) that they are inconsiste nt systems. 3. Show 3, 4 of row reduced echelon forms Find.

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26 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.

27 Definiton: A vector space V in R is the nonempty set V satisfying the followings If u, v V and a R, then u v and au V. u v v u (commutative) u ( v w) ( u v) w, u,v,wv u u (a unique zero element) u ( u) (a uique additive inverse) ( a b) u ( ab) u u a( bu), a,b R and u V. u (here is identity scalar) a( u v) au av au bu (distributive ) Eamples. V {}, P n [] F[a,b] (space of V n R V polynomials), (function - space on interval[a, b]), R mn (space of matrices),

28 Linear Dependence Define a linear combination of c u c u... c n u n, nonzero vectors where u i V, c i R, i,,3,... If the equation c u has only trivial solution for all c Otherwise (if any c c u... c i n u n is nonzero), i, then u, u,... u n are linealy independent. then they are linearly dependent.

29 Page 97 () Eample: A= RankA= because the first two row vectors are linearly independent.

30 3.. to equivalent row is 3 : ranka A Eample

31 Page 98 (3)

32 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. Eamples : dim R n n, dim R mn, dim P [ ] n, dim F[ a, b], n mn dimc n [ a, b]

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

34 Pages 3-33a ~ Let ranka ranka. Then the system () i. ii. has infinitely many solutions if r n. has a unique solution if ad only if r n. Advanced Engineering Mathematics by Erwin Kreyszig Copyright 7 John Wiley & Sons, Inc. All rights reserved.

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37 Page 35 () solution. ) ( : free parametric n-r Two, r n m Eample

39 .,, 3 where, are solution particular with a solutions All : R c c Eample T h T h

40 Determinant Page 38a

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47 Inverse of a matri using Gauss-Jordan Elimination. Eample : Find inverse of A 38a if its inverse eists A Do some elementary row operations to [ A, I] and get [ I, A ].

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50 Eamples for inner product spaces

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

55 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 } ), ( : { }., ) ( : { be a : Let

Applied Mathematics Course Contents

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