Class Note for MATH 215 with Professor Dostert at UA
Class Note for MATH 215 with Professor Dostert at UA
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Date Created: 02/06/15
THE UNIVERSITY a OF ARIZONA Math 215 Introduction to Linear Algebra Section 22 Direct Methods for Solving Linear Systems Paul Dostert September 26 2008 Ac Matrices and Row Echelon Form For a linear system the coefficient matrix contains the coefficients of the variables and the augemented matrix is the same but with an extra column containing the constant terms We usually denote the coefficient matrix by A the constant vector by b and the variable matrix as ac The augmented matrix is denoted by Ab A matrix is in row echelon form if a Any rows with all zeros are at the bottom b The leading entry in each nonzero row is in a column to the left of the rows below it Ex The following are in row echelon form 1 2 3 4 4 5 17 1 1 0 2 0 8 11 6 0 9 3 0 3 3 2 7 0 0 1 2 0 0 4 0 0 9 11 0 0 0 6 0 0 0 0 At Elementary Row Ops amp Gaussian Elim In order to transform a matrix to row echelon form we use elementary row operations defined by any of the following a Interchanging two rows b Multiplying any row by a constant c Adding a multiple of any row to another row The process of reducing to row echelon form is called row reduction Two matrices are row equivalent if elementary row operations can be used to transform from one to the other They are row equivalent iff they can be reduced to the same row echelon form To solve a linear system we can use Gaussian Elimination G E We 1 Write the augmented system 2 Reduce to row echelon form 3 Solve by back substitution A Gaussian Elimination When reducing to row echelon form you should attempt to zero out each lower diagonal entry starting with the first column until finished To do this you should always move a row with a nonzero entry on the diagonal Ex Solve the following systems by GE 2 3 5 2 1 4 2 1 1 0 4 1 7 8 4 2 2 2 1 0 1 4 2 1 1 If we apply GE to a system with inifinitely many solutions then we have the leading variables in terms of the other free variables which can be any value In a consistent system the free variables are the non leading variables For yzwx 1 z2w 2 x 8 The leading variables would be y z and c The free variable would be w Ah Rank The rank of a matrix is the number of nonzero rows in its row echelon form Thm Let A be the coefficient matrix of a system with n variables If the system is consistent then number offree variables n rank A Note We will learn many other ways to determine and use the rank later A matrix is in reduced row echelon form rre if it satisfies 1 It is in row echelon form 2 The leading entry in each nonzero row is 1 3 Each column containing a 1 has zero elsewhere The following are in reduced row echelon form 1 O 7 O O 00lk Ol O l lOO 00lk Ol O OllkN l lOO Ol OO OOl O Ala GaussJordan Elimination GaussJordan Elimination GJE is a process to solve a linear system using the rref Its steps are 1 Write augmented matrix 2 Reduce to rref using elementary row operations 3 If the resulting system is consistent solve for leading variables in terms of free variables Ex Solve the following systems by GJE 2 3 5 2 1 4 2 1 1 0 4 1 7 8 4 2 2 2 1 0 1 4 2 1 1 A Homogeneous Systems and Examples A system of linear equations is homogeneous if each constant term is zero if the RHS is zero Thm If A0 is a system of m equations with n variables with m lt n then the system has infinitely many solutions Ex Find the intersection of the planes c y z 0 2x y z 0 and x y z 0 Ex Find the intersection of the planes c y z 0 and c 2y z 4 Ex A bakery has 3 different sizes of cupcakes mini for 1 regular for 2 and monster for 4 The number of monster and regular cupcakes adds to twice the number of mini cupcakes If the company has 300 cupcakes worth 600 then how many of each cupcake do they have A Matlab We re going to jump ahead and bit an learn how to write and solve linear systems as matrices Suppose we have 4 6 2 0 3 8 11 4 1 4 2 11 In Matlab we need to write the augmented matrix as the coefficient matrix and a column vector containing the constants the RHS vector To write a matrix or vector we use brackets A comma separates entries in a row and a semicolon indicates a move to a new row Generally we refer to the matrix as A and the RHS vector as b The command c Ab means solve Ax b for m using Gaussian Elimination We write A 46 2 381 1 14 2 b 041 1 X A b This will only work correctly for uniquely solveable systems Othenvise it will give you a possible solution if there are infinitely many solutions or a close solution if there are no solutions
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