Special Two-Semester Linear Algebra Course (Fall 2012 and Spring 2013) The first semester will concentrate on basic matrix skills as described in MA 205, and the student should have one semester of calculus. The second semester will build on the first semester and will be a little more theoretical. There will be ample time to do some significant applications with SVD, FFT, wavelets and BVP. General inner product spaces will be studied as is indicated the tentative course number MA 405_493_591 and description given below. MA 205: Elements of Matrix Computations Time and Place: 1:30 pm-2:45 pm, TuTh, SAS????, Fall Semester of 2012 Instructor: Robert E. White, Professor of Mathematics, NCSU, SAS 3140, 515-7478, white@math.ncsu.edu Prerequisites: C or better in first semester of calculus (MA 121 or 131 or 141) CATALOG DESCRIPTION: (three-credit course; credit allowed for only one of MA 205, 305 and 405) Complex numbers and Euler's formula. Vectors in 2-D and 3-D, lines, planes, vector products and determinants. Vectors in n-d, matrices and matrix products. Algebraic systems, row operations, inverse matrices and LU factors. Least squares, underdetermined systems and null and column spaces. Applications to linear systems of differential equations and/or to visualization with image filters. Emphasis is on by-hand computations, but it is to include applications and computing tools. MA 405_493_591: Linear Matrix and Differential Operators Time and Place: 1:30 pm-2:45 pm, TuTh, SAS????, Spring Semester of 2013 Instructor: Robert E. White, Professor of Mathematics, NCSU, SAS 3140, 515-7478, white@math.ncsu.edu Prerequisites: C or better in MA 205 or equivalent POSSIBLE CATALOG DESCRIPTION: (????) Builds on matrix calculations for solving Ax = d where A is mxn. General inner product spaces, basis and four fundamental spaces. Orthogonal vectors and functions, Gram-Schmidt and QR factors. Eigenvalues for linear operators, linear IVP, adjoint operators and Sturm-Liouville BVP. SVD, least squares and image compression. Haar transform and introduction to wavelets. Fourier transform, filters and Poisson solvers.
WHO SHOULD TAKE THE FIRST SEMESTER? This course will serve to "diversify" math course work and to provide "math-on-time" for first year students in engineering and the sciences. Some returning or transfer students may need a review of 2-D and 3-D vectors and matrices, basic calculus and need to learn a relevant computing tool such as MATLAB. In particular, the course will provide critical assistance to the student in the following four areas: 1. Discrete models require matrix algebra and computing skills. This course will better prepare students with one semester of calculus to study discrete and statistical modeling, multivariable calculus (MA 242), differential equations (MA 341) and additional matrix (linear or numerical) algebra. 2. For students of physical sciences and engineering, this course will provide math-on-time so that they will have matrix computation skills and linear ordinary differential equations at the beginning of the second year. 3. For students in the biological and management sciences, this course will provide additional matrix oriented mathematics without having to take a three-semester calculus sequence. The first part of the multivariable calculus course (on vectors in 2-D and 3-D, lines, planes, vector products and determinants) will be covered in the first quarter in MA 205. 4. For students in liberal arts the coupling of one semester of calculus with this elementary matrix course will provide an excellent technical perspective on modern mathematics, applications and computations. A more detailed understanding of the sensational evolution of computer hardware and software will be gained. This course is newly approved for the General Education Program (GEP effective July 2009), which is an enhanced version of GER. WHO SHOULD TAKE THE SECOND SEMESTER? Students should have basic skills in matrix computations as in MA 205 or equivalent, which will be quickly reviewed in the first four or five lectures. Although careful attention to definitions and mathematical analysis will be stressed, there will be ample time for applications that utilize computing tools. MATLAB will be used to illustrate these important topics. Students in the mathematical and physical sciences and engineering will find this course to be very useful in subsequent studies. In the next 20 lectures linear operations on Euclidean and function spaces will be studied. Here bases, orthogonal and eigenvalue/eigenvectors and similarity will be carefully studied. There will be more time for examples and for careful analysis of the fundamental concepts. The next 15 lectures will include about three lectures on each of the following: SVD and image compression, Haar transform and introduction to wavelets, FFT used in filters and Poisson solvers, Sturm-Liouville BVP and eigenfunctions, and Previews of related topics. The last three lectures will be previews of numerical linear algebra (MA 428, MA 580), general inner product spaces (MA 520, MA 523) and Hilbert spaces (MA 515).
Elements of Matrix Computations: Possible 43 50-minute Lectures Lecture Sections in R. E. White Topic 1 1.1 Complex numbers and vectors 2 1.2 Complex valued functions 3 1.2 Euler s formula 4 1.3 Vectors in 2D 5 1.4 Dot product and work 6 1.5 Lines and curves in 2D 7 2.1 Vectors in 3D 8 2.2 Cross and box products 9 2.3 Lines and curves in 3D 10 2.4 Planes in 3D 11 Test one 12 3.1 Matrix models 13 3.2 Matrix products 14 3.3 Special cases for Ax = d 15 3.4 Row operations 16 3.4 Gauss elimination 17 3.5 Inverse matrices 18 3.6 LU factorization 19 3.7 Determinants and Cramer s rule 20 Test two 21 4.1 Curve fitting to data 22 4.2 Normal equations 23 4.3 Multilinear data fitting 24 4.4 Parameter identification 25 5.1 Multiple solutions 26 5.2 Row echelon form 27 5.2 Subspaces 28 5.3 Nullspaces and equilibrium equations 29 Test three 30 6.1 First order linear DE 31 6.2 Second order linear DE 32 6.3 Homogeneous and complex solutions 33 6.4 Nonhomogeneous solutions 34 6.4 Solution of IVP 35 6.5 System form of second order linear DE 36 7.1 Solution of x = Ax by elimination 37 7.2 Real eigenvalues and eigenvectors 38 7.2 Eigenvalues 39 7.3 Solution of x = Ax 40 Test four 41 7.3 Solution of x = Ax + f(t) 42 Review Comprehensive final exam 43 Review Comprehensive final exam
Linear Operators: Possible 43 50-minute Lectures Lecture Sections in G. Strang Topic 1 1.1-1.3 Matrix products 2 2.1-2.7,5.3 Solution of Ax = d where A is nxn, row operations 3 4.3 Solution of Ax = d where A is mxn and m>n, normal eq. 4 3.3 Solution of Ax = d where A is mxn and m<n, REF 5 3.1, 8.5, 10.2 Inner product spaces 6 3.2, 7.1 Nullspaces of linear operators (matrix and differential) 7 3.3a Rank of a matrix 8 3.3b RREF 9 3.4 Set of all solutions for linear operators 10 Test one 11 3.5a Basis in R n 12 3.5b Basis for nullspace of matrix and differential operators 13 3.6 Four fundamental spaces 14 4.1 Orthogonal vectors 15 4.2 Projection 16 4.3 Least squares 17 4.4a Gram-Schmidt 18 4.4b QR factorization 19 9.1, 9.2 Ill-conditioned problems 20 Test two 21 6.1 Eigenvectors 22 6.2 Normal matrices 23 6.3 Initial and boundary value problems 24 6.4-6.6 Symmetric positive definite matrices 25 6.7a, 7.3 SVD, least squares 26 6.7b Image compression 27 7.1 Linear transforms 28 7.2 Haar and Fourier transforms 29 notes Introduction to wavelets 30 Test three 31 10.1, 10.2 Fourier transform and FFT 32 10.3a, notes Image filters 33 10.3b, notes Poisson solver 34 8.1 Matrices in engineering: diffusion models 35 8.2 Matrices in engineering: equilibrium equations 36 8.5 Fourier series 37 S-L notes Sturm-Liouville BVP 38 S-L notes Real eigenvalues 39 S-L notes Orthogonal eigenfunctions 40 Test Four 41 8.5, notes Hilbert space examples (leading to MA 515) 42 9.1, 9.2, 9.3 Numerical linear algebra (leading to MA 428, 580) 43 notes General adjoint operators (leading to MA 520, 523)
What is the difference between MA 205, 305 and 405? This is a mixture of fact and opinion by Robert E. White The current (fall 2011) course descriptions are listed below. They indicate MA 405 is to be more theoretical than MA 305, with MA 405 having prerequisite of calculus two (MA 241) and recommended math foundations (MA 225) and ordinary differential equations (MA 341). The traditional introduction to vectors, lines and planes and vector products 2D and 3D is given in calculus three (MA 242), which has been dropped as a prerequisite for MA 305 and MA 405. The first quarter of MA 205 includes these topics so as to give geometric insight and motivation for their generalizations to higher dimension. Many disciplines are becoming more quantitative, and they require discrete models where matrix structures and computations play an integral role. In the mathematical sciences this evolution in the last 30 years has been sensational. But, our core undergraduate course offerings have not been significantly updated! Math/engineering/science majors need to have two semesters of matrix/linear algebra. MA 205 should be taken after one semester of calculus, and it should be a methods oriented course as are our calculus sequences. The proposed course, MA 405_493_591, is an attempt to provide some undergraduate and graduate students with topics in linear algebra that are relevant to their interests. MA 205 Elements of Matrix Computations UNITS: 3 Prerequisite: C- in MA 121, 131, or 141 Complex numbers and Euler's formula. Vectors in 2-D and 3-D, lines, planes, vector products and determinants. Vectors in n-d, matrices and matrix products. Algebraic systems, row operations, inverse matrices and LU factors. Least squares, undetermined systems and null and column spaces. Applications to linear systems of differential equations and/or to visualization and image filters. Emphasis is on by-hand computations, but it is to include applications and computing tools. Students cannot receive credit for more than one of MA 205, MA 305, or MA 405. MA 305 Introductory Linear Algebra and Matrices UNITS: 3 - Offered in Fall Spring Summer Prerequisite: MA 241 or MA 231 with MA 132 The course is an elementary introduction to matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, Euclidean vector spaces, determinants, eigenvalues and eigenvectors, linear transformations, similarity, and applications such as numerical solutions of equations and computer graphics. Compares with MA 405 Introductory Linear Algebra, more emphasis is placed on methods and calculations,. Credit is not allowed for both MA 305 and MA 405. MA 405 Introduction to Linear Algebra UNITS: 3 - Offered in Fall Spring Summer Prerequisite: MA 241 (MA 225 recommended); Corequisite: MA 341 is recommended This course offers a rigorous treatment of linear algebra, including systems of linear equations, matrices, determinants, abstract vector spaces, bases, linear independence, spanning sets, linear transformations, eigenvalues and eigenvectors, similarity, inner product spaces, orthogonality and orthogonal bases, factorization of matrices. Compared with MA 305 Introductory Linear Algebra, more emphasis is placed on theory and proofs. MA 225 is recommended as a prerequisite. Credit is not allowed for both MA 305 and MA 405.