Class Notes (835,929)
Canada (509,507)
Mathematics (1,919)
MATH 136 (168)
Lecture

lect136_3_w13.pdf

5 Pages
82 Views
Unlock Document

Department
Mathematics
Course
MATH 136
Professor
Robert Andre
Semester
Spring

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
More Less

Related notes for MATH 136

Log In


OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Sign up

Join to view


OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.

Reset Password

Please enter below the email address you registered with and we will send you a link to reset your password.

Add your courses

Get notes from the top students in your class.


Submit