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seg2018_08-1 - ERG 2018 Advanced Engineering Mathematics...

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Unformatted text preview: ERG 2018 Advanced Engineering Mathematics Lecture Notes Dr. K.P. Lam Sept, 2008 1 Major topics Calculus Calculus of multiple variables Linear Linear algebra and vector space Linear Linear transformation, matrix decomposition, eigen-analysis eigen- 2 Introduction Motivations Motivations • To apply mathematics for practical applications in engineering, economics, finance, and others • To consider practical cases and examples where the major topics can be applied 3 Introduction example 1-- Physical 1-system modeling Laplace Laplace equation for the temperature inside a rectangular container • Partial derivatives and ordinary derivatives, gradient and curvature, first and second derivatives • Laplace operator on temperature • Formation of linear equation set • Other domains (electromagnetic field, biomedicine, weather forecasting) 4 Introduction example 2 – Financial modeling and forecasting TimeTime-series modeling for stock prices • Stock price today may have a linear relationship with the prices yesterday and the day before • Formation of a linear equation model • Using historical data to obtain the model • Using the model for forecasting 5 Linear equations and matrices Two Two linear equations • Using the row and column indexes (to prepare for generalization) • Geometrical interpretations in a 2-D plane 2(intersection, parallel lines, overlap) • Matrix formulation ( Ax = y ) Generalization nGeneralization to n-dimensional case 6 Solving linear equations Cramer’s Cramer’s rule • 2-dimensional and 3-dimensional cases 3but becomes tedious for high dimensions • Matrix determinants Gauss Gauss elimination • Good for a general n-dimensional system n• Two-step procedure: forward elimination Twoand backward substitution 7 Backward substitution Consider Consider special cases for A: L and U as lower and upper triangular matrices Solving Solving Ux=y is much simpler (why?) • 2-dimensional: x2 easily obtained, then x1 • 3-dimensional: x3 easily obtained, then x2 and x1 • n-dimensional? 8 Forward elimination How How should we reduce a general Ax=y to Ux = z? Use Use elementary row operations on the augmented matrix [A | y] and change it to [U | z] in a columnwise manner Example: 2Example: 2-dimensional case How How can we generalize to a procedure for general n dimensions? 9 GaussGauss-Jordan and matrix inverse Finding Finding x by the reduction of [A| y] to [I | z] using elementary row operations – GaussGauss-Jordan • 3-dimensional example Finding Finding x for Ax=y using x=A-1y. Can we find matrix inverse using similar GaussGaussJordan procedure? 10 Vectors Basic Basic vector operations Norm, Norm, dot product, and angle for nnspace Orthogonality Orthogonality Schwarz Schwarz inequality • The relationship between |u.v| and the |u.v| norms • Triangle inequality 11 11 Vector space Definition Definition of a vector space. What are the main properties? Subspace Subspace and span Linear Linear dependence (LD) of a set of vectors • Linear combination of vectors 12 ...
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