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# LinearModelingforEngineers MATH290

Drexel

GPA 3.63

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This 20 page Class Notes was uploaded by Moshe Swift III on Wednesday September 23, 2015. The Class Notes belongs to MATH290 at Drexel University taught by DavidKimsey in Fall. Since its upload, it has received 16 views. For similar materials see /class/212275/math290-drexel-university in Mathematics (M) at Drexel University.

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Date Created: 09/23/15

Math 290 Linear Modeling for Engineers Instructor Minnie Catral Text Linear Algebra and Its Applications 3iOI ed Update by David Lay Have you ever used Matlab as a computing tool 1 Yes 2 No 11 Systems of Equations System of 2 equations in 2 unknowns L123 bly 612 523 Cl C2 Three possibilities 1 The lines intersect 2 The lines coincide 3 The lines are parallel More generally System of m equations in n unknowns a111a122a1nn bl a211a222a2nn 52 n 4m4n Ami m L Lomle T Lomsz T T ujmnwn um A system is consistent if it has at least one solution Othenlvise it is said to be inconsistent FACT A system of linear equations has either 1 exactly one solution or 2 infinitely many solutions or 3 no solutions The solution set is the set of all possible solutions of a linear system Two systems are equivalent if they have the same solution set STRATEGY FOR SOLVING A SYSTEM Replace a system with an equivalent system that is easier to solve EXAMPLE 1 22 2131 3132 Matrix Notation 13121L 2 3 1 25131 3132 1 2 Elementary Row Operations 1 Interchange Interchange two rows 2 Scaling Multiply a row by a nonzero constant 3 Replacement Replace a row by the sum of itself and a multiple of another row Row equivalent Matrices Two matrices where one matrix can be transformed Into the othermatrlx by a sequence of elementary row operations Notation A N B Fact about Row Equivalence If the augmented matrices of two linear systems are row equivalent then the two systems have the same solution set 8 39 EXAMPLE 2562 53933 2232 83 43131 I 5562 I 9563 931 OO Two Fundamental Questions Existence and Uniqueness 1 Is the system consistent ie does a solution exist 2 If a solution exists is it unique ie is there one and only one solution 10 EXAMPLE Is this system consistent 5x1 6133 2 Julj 3132 22 7182 9x3 8 l O 11 EXAMPLE For what values of h will the following system be consistent 4 h 3131 9132 2z1 6132 12 Determine the value of h such that the matrix is the augmented matrix of a consistent system 3 00 P 90 gt 4 a a 3 3 3 5 13 12 Row Reduction and Echelon Forms A matrix is in echelon form or row echelon form if 1 All nonzero rows are above any rows of all zeros 2 Each leading entry ie leftmost nonzero entry of a row is in a column to the right of the leading entry of the row above it 3 All entries in a column below a leading entry are zero 14 Examples Determine which matrix is in echelon form o 3 4 a 1 2 1 o o o 391 1 139 b 0 0 1 0 1 0 390 1 5 2 1 O Oquot C O O O 3 6 3 9 O O O O O 2 4 O O O O O O O A matrix is in echelon form or row echelon form if 1 All nonzero rows are above any rows of all zeros 2 Each leading entry ie leftmost nonzero entry of a row is in a column to the right of the leading entry of the row above it 3 All entries in a column below a leading entry are zero A matrix is in reduced echelon form or row reduced echelon form if in addition to conditions 123 above 4 The leading entry in each nonzero row is 1 5 Each leading 1 is the only nonzero entry in its column 16 O 1 5 2 1 O O O O O 3 6 3 9 Example 0 0 o 0 c 0 0 is in echelon form but not In reduced echelon form 17 Uniqueness of the Reduced Echelon Form Each matrix is row equivalent to one and only one reduced echelon matrix 18 Important Terms pivot position a position of a leading entry in an echelon form of the matrix pivot a nonzero number that either is used in a pivot position to create 0 s or is changed into a leading 1 which in turn is used to create 0 s pivot column a column that contains a pivot position 19 Example nu nu 1 n 40 O O O 3 6 3 9 OO 00 00 OO 00 20

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