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Lecture 10

# Lecture 10.pdf

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School
University of Waterloo
Department
Mathematics
Course
MATH 136
Professor
Robert Sproule
Semester
Winter

Description
Monday, January 27 − Lecture 10 : Gauss-Jordan elimination algorithm for solving systems of linear equations. Concepts: 1. Gauss-Jordan elimination algorithm for solving systems of linear equations. 2. consistent and inconsistent system. 3. ERO’s We see that solving a system of linear equations which is in RREF is a relatively easy task. However most systems we encounter are not in RREF. We now look at a technique where we transform such systems in RREF without changing the solution. 10.1 Definition − The Gauss-Jordan elimination is an algorithm which transforms a system of linear equations onto an RREF system in such a way that the resulting system has the same solution as the original system of equations. These changes are done by performing 3 "Elementary row operations" (ERO’s) on the augmented matrix associated to the given system : - ERO of type I : Interchange two rows of the system. We denote this row operation by P ijmeaning we "permute" or exchange the i and j rows. th - ERO of type II : Multiply a row of the system by a non-zero real number. We denote this by cR, mianing we multiply the row i with the scalar c. - ERO of type III : Replace the i row R with the sum of R and the scalar multiple of i i row R j. We denote this by R + cR. i j Note: We could actually apply the above operations to the system of equations and referring to these as the 3 “equation-operations”: - “ P ij, interchange the order of two equations, - “cE”,imultiply both sides of equation E by a non-zeio scalar c, - “E +icE”, rejlace equation E, by the sui of the equation E and the equation i obtained by multiply both sides of the equation E jby c. Anyone of these 3 operations does not change the solution of the system. The elementary row operations essentially describe the operations performed above in solving the system of linear equations. 10.1.1 Note − Performing the elementary operations on a system always results in a system which has the same solution as the system we started off with. 10.1.2 Remark − Any ERO can be "undone" by an ERO: - To "undo" P apijy P ij - To "undo" cR appiy (1/c)R. i - To "undo" R + ci applyjR + (−c)Ri j. Verify this with a simple matrix. 10.1.3 Remark − For some there may be some confusion about the interpretation given to an ERO such as, for example, 2R + R . 3 5 - The ERO “2R + R 3 is t5 be interpreted in the same ways as “R + 2R ”. It 5 3 means “Replace row 5 with 2 times row 3” - Just to avoid confusion some may prefer to state the ERO as: R → R + 2R 5 5 3 which more clearly states “replace row 5 with 2 times row 3”. - But do not write “R → 23 + R ”,3which5will surely confuse the reader. 10.2 General strategy for solving linear systems by using the Gauss-Jordan elimination. System of linear equations → Obtain the augmented coefficient matrix [A|b] → Apply ERO’s to transform this matrix matrix [A|b] in RREF, [A RREF | d] → Transform back into the system of linear equations of [A RREF| d] → Isolate the basic variables. → Write out the solution set in scalar form. → Write out the solution in “vector equation form”. 10.2.1 Note − If the operations are done accurately the solution to the system of linear equation [A RREF | d] is identical
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