PPT-Linear Systems and Matrices

Algebra 2 Chapter 3 This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson R Boswell L Kanold T D amp Stiff L 2011 Holt McDougal Some examples and diagrams are taken from the textbook. In particular they are useful for compactly representing and discussing the linear programming problem Maximize subject to i j This appendix reviews several properties of vectors and matrices that are especially relevant to this problem We shoul e Ax where is vector is a linear function of ie By where is then is a linear function of and By BA so matrix multiplication corresponds to composition of linear functions ie linear functions of linear functions of some variables Linear Equations a 12 22 a a mn is an arbitrary matrix Rescaling The simplest types of linear transformations are rescaling maps Consider the map on corresponding to the matrix 2 0 0 3 That is 7 2 0 0 3 00 brPage 2br Shears The next simplest type of linear transfo Hermitian skewHermitian and unitary matriceseigenvalues and eigenvectors diagonalisation of matrices CayleyHamilton Theorem Calculus Functions of single variable limit continuity and differentiability Mean value theorems Indeterminate forms and LHos Elementary Row Operations To solve the linear system algebraically these steps could be used 11 12 z8 All of the following operations yield a system which is equivalent to the original Equivalent systems have the same solution Interchange equatio Matrices. Definition: A matrix is a rectangular array of numbers or symbolic elements. In many applications, the rows of a matrix will represent individuals cases (people, items, plants, animals,...) and columns will represent attributes or characteristics. Lectures 1-2. David Woodruff. IBM Almaden. Massive data sets. Examples. Internet traffic logs. Financial data. etc.. Algorithms. Want nearly linear time or less . Usually at the cost of a randomized approximation. Matrix Multiplication. Matrix multiplication is defined differently than matrix addition. The matrices need not be of the same dimension. Multiplication of the elements will involve both multiplication and addition. Niebles. . and Ranjay Krishna. Stanford Vision and Learning . Lab. 10/2/17. 1. Another, very in-depth linear algebra review from CS229 is available here:. http://cs229.stanford.edu/section/cs229-linalg.pdf. Chapter 3.8. Square Matrix. Although a matrix may have any number of rows and columns, . square matrices. have properties that we can use to solve systems of equations. A square matrix is one of the form . Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: . 29. th. August 2015. Introduction. A matrix (plural: matrices) is . simply an ‘array’ of numbers. , e.g.. But the power of matrices comes from being able to multiply matrices by vectors and matrices by matrices and ‘invert’ them: we can:. What is a matrix?. A Matrix is just rectangular arrays of items. A typical . matrix . is . a rectangular array of numbers arranged in rows and columns.. Sizing a matrix. By convention matrices are “sized” using the number of rows (m) by number of columns (n).. A cofactor matrix . C. of a matrix . A. is the square matrix of the same order as . A. in which each element a. ij. is replaced by its cofactor c. ij. . . Example:. If. The cofactor C of A is. Matrices - Operations. Rotation of coordinates -the rotation matrixStokes Parameters and unpolarizedlight1916 -20041819 -1903Hans Mueller1900 -1965yyxyEEEElinear arbitrary anglepolarization right or left circularpolarizati