Class Note for MATH 410 with Professor Dostert at UA
Class Note for MATH 410 with Professor Dostert at UA
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Date Created: 02/06/15
THE UNIVERSITY Q OF ARIZONA Math 410 Matrix Analysis Section 11 Solution of Linear Systems Section 12 Matrices and Vectors Paul Dostert December 28 2008 112 A Linear Systems A linear equation in n variables x1 written as xn is an equation that can be I L axan ZaZxiaxb z391 where az are coefficients and b is a constant term A solution to a linear equation is a set of xz often written as a vector such that the set 36239 satisfies the equation A system of linear equations is a finite set of linear equations The solution to the system must satisfy each equation in the system We call the solutions to the system the solution set Two linear systems are called equivalent if they have the same solution sets A system of linear equations with real coefficients has either a a unique solution a consistent system b infinitely many solutions also a consistent system c no solutions an inconsistent system An Example of a Linear System Consider the system acyz 1 xy z 2 2x yz 8 This is linear since each unknown appears only as a first power How are 1 each of the following related to the previous system of equations xyz 1 xyz 1 2x2y 3 7 2x2y 2z 4 2x yz 8 2x yz 8 At Properties of Linear Systems Linear System Operation 1 We can add a multiple of one equation to another equation and not change the solution to the system Two systems of equations are equivalent if they have the same solution By using operation 1 we can reduce the equation on the previous slide to xyz 3y z 2z l Ql l We call this form triangular since the last equation involves only 1 unknown the 2nd to last has only 2 etc We can solve a triangular system by back substitution which involves solving the last equation first and substituting it back into the second to last equation and so on A Matrices A matrix is a rectangular array of numbers called the entries also called the elements of the matrix The following are all matrices 1 1 1 0 1 1 1 1 1 1 021 2 12 10 4 0 rib5 11761207lli The size of a matrix is the number of rows and columns A general m gtlt n matrix A can be written as all Q12 aln Q21 a22 a2n A aml am2 amn The entries of A are denoted by aZj where z is the row number and j is the column number Note A vector is also a matrix A row vector is a 1 gtlt n matrix called a row matrix Similarly a column vector is a m gtlt 1 matrix called a column matrix All Matrix Definitions 2 Two matrices are equal if they have the same size and the same entries Consider a general linear system of m equations and n unknowns 011901 012902 39 quot a1n90n 517 0021901 a222 39 quot Cb2n90n 527 anlxl an2x2 39 39 39 annxn We define A am as the m gtlt n coefficient matrix and the column vectors corresponding to the unknowns and right hand side as 301 b1 X f andb vn bn Ex Write the following system in matrix vector form acyz 1 acy z 2x yz 8 At Matrix Operations Consider 1 1 1 1 0 0 1 2 2 1 A 1 1 1 7 2 1 0 7 3 2 1 0 2 1 2 0 0 7T 61 2 sin1 3 Matrix addition is defined componentwise between matrices of the same size In other words if A and B are m gtlt n matrices then the ijth element of A Bis given aij EX Add A and B Add B and C Scalar multiplication takes a scalar c and an m gtlt n matrix A and computes the m gtlt n matrix given by multiplying each entry of A by 0 Ex Calculate 50 Calculate A Ac Matrix Multiplication The product of two matrices is NOT done componentwise If A is an m gtlt n matrix and B is an n gtlt 7quot matrix then C AB is an m gtlt 7quot matrix given by n Cij ailblj 39 ambnj E aikbkj k1 The ijth entry of C is ith row of A multiplied by the jth column of B Note A and B do not need to have the same size but the number of rows of A must be the same as the number of columns of B In terms of sizes we haveZannxrmgtlt7quot IfDiandE12 then lt1gtlt 2gt lt2lt1gt lt1lt3gt lt2lt7gt 0 17 DE lt3lt 2gtlt4gtlt1gt lt3gtlt3gt lt4gtlt7gt l 2 37 l 39 Ex Calculate AB AC and CA from the previous slide Ac Types of Matrices A zero matrix is an m gtlt n matrix of all zeros Example of zero matrix always called 0 O 8 8 An identity matrix is an n gtlt n square matrix containing ones on the main diagonal the am 139 17 n entries and zeros on all of the off diagonal entries the aim z 7 j entries Example of identity matrix always called I I OOH Ol O woo The identity matrix and zero matrix are specific forms of a diagonal matrix which contains all zeros on the off diagonals 1 D 0 0 0 0 Example of diagonal matrix 2 0 0 4 A Matlab Matrix Operations Most usual operations work as expected in Matlab It will return an error if you do something incorrectly like add two matrices of different sizes An identity matrix is formed using the eye command A zero matrix can be formed using zeros Try the following A 2345 B 2 103 C eye22 D zeros22 A2B B C Block matrix operations work exactly as expected as well To define a 4 gtlt 4 block matrix A B Ge l C Di we simply define a 2 gtlt 2 matrix with matrices as elements G AB CD All Matlab Matrix Multiplication Matrix multiplication brings up an interesting issue in Matlab Let us say we wish to find Ak as k gt 00 for Al 3 Everything works as expected in Matlab A 012112 AAZ AA8 AA IG What does it look like this is approaching We find any l a a l Similarly we can multiply two different matrices B01 1 1 12 BA AB What will happen when we try AB We ll get an error since AB is not well de ned A Matlab Matrix Multiplication 2 Matlab also has the ability to apply operations to entry individually If we use the command before an operator then each operation is defined componentwise Note that this is ONLY defined for matrices of the same size Using componentwise arithmetic we have A i 0k 12 A39k1 lt12gt So we can try Aquot2 Aquot8 Aquot16 We find A 7 0 0 kIEEOA k1 0 Similarly we can multiply A componentwise to a matrix B of the same size B01 1 1 AB BA Note A gtllt B B gtllt A but in general AB 7 BA for A and B of the same size
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