seg2018_08-1

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online