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# MATH-M303 Section 1.1/part of 1.2 Notes MATH-M 303

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This 2 page Class Notes was uploaded by Kathryn Brinser on Wednesday August 24, 2016. The Class Notes belongs to MATH-M 303 at Indiana University taught by Keenan Kidwell in Summer 2016. Since its upload, it has received 12 views. For similar materials see Linear Algebra for Undergraduate in Mathematics at Indiana University.

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Date Created: 08/24/16

M303 Section 1.1 Notes- Solving Systems of Linear Equations 8-22-16 Linear equation in variables1???? ,…,???? - equation of form1 1???? + 2 2 + ⋯+ ???? ???? ???? ????, where coefficients ???? and ???? are real numbers Linear system- collection of 1 or more linear equations in the same variables ???? ,…,???? 1 ???? o Ex. 3????1− 2???? 2 6 9????2= 10 o Ex. ???? − ???? + ???? = 2 1 2 3 ????1+ ???? 3 0 Solution of system in 1 ,…,????????- ordered ????-tuple 1 ,…,????????)of real numbers ????????, which, when substituted for ???? , result in all equations being true ???? o Ex. Can check by inspection that 1,−2,−1 is solution to second equation above o Solution set- set of all possible solutions to linear system Two systems equivalent when they have same solution set For system of 2 equations in 2 variabl1s ???? an2 ???? , each equation defines a li1 2in ???? ???? plane o Solution is ordered pair/tuple in the plane that lies at intersection of the lines o Recall for all linear systems of any number of variables, 3 possibilities of intersection: One point (unique solution) All points (parallel identical lines, infinitely many solutions) No points (parallel, non-intersecting, no solution) o System is consistent if it has solutions; inconsistent otherwise ???? × ???? (“???? by ????”) matrix- rectangular array of numbers with ???? rows and ???? columns: ???? ⋯ ???? 11 1???? o ???? = [[ ⋮ ⋱ ⋮ ]] ???? ????1 ⋯ ???? ???????? o Entry of ???? in ????row and ????????ℎ column denoted ???? , called ????????ℎentry ???????? o Ex. There are 2 matrices associated with a linear system given the system: ????1− 2???? 2 ???? =30 { 2???? − 8???? = 8 2 3 5????1− 5???? =310 Coefficient matrix- 3 × 3 using left sides of equations 1 −2 1 ???? = [0 2 −8 ]where column 1 is ???? , column 2 is ???? , column 3 is ???? 1 2 3 5 0 −5 Augmented matrix- to keep track of right sides 1 −2 1 0 ???? ???? = [0 2 −8 8 ]where left of bar is ???? (coeff.), right of bar is vector ???? (solutions) 5 0 −5 10 Can be used to write down corresponding system for given augmented matrix Want to transform matrix of a given system into “simpler” matrix for an equivalent system, such that solutions can be found by inspection o Simple matrix- system whose augment matrix is the identity matrix on the left and ???? o Ex. [1 0 0 −| ] 0 1 0 2 ????1= −1, ???? 2 2, ???? i3 a free variable Solution set is (−1,2,????) where ???? ???? ℝ Elementary row operations- to obtain equivalent systems o Row replacement- replace some row ???? with row ???? + ????(row ????′) for some ???? ???? ℝ; adding a multiple of another row to the row in question; ???? → ???? + ???????? ′ ′ ???? ???? ???? o Row swap- ???? ???? ???? ???? o Scaling- multiply row by nonzero number ????;???????? → ???????????? o DON’T FORGET to do operations to solutions in matrix too o Ex. Find all solutions to the following system: ????1− 2????2+ ???? 3 0 2????2− 8???? 3 8 5????1− 5????3= 10 Make zeroes below diagonal entries 1 −2 1 0 ????3→ ???? 3 5???? 1 [0 2 −8 | 8 ] 0 10 −10 10 1 1 −2 1 0 ????2→ ???? 2 [0 1 −4 | 4 ] 2 0 10 −10 10 1 −2 1 0 ???? → ???? − 10???? [ | ] 3 3 2 0 1 −4 4 0 0 30 −30 1 1 −2 1 0 ????3→ 30????3 [0 1 −4 4 ] 0 0 1 −1 Can see 3 = −1, but still need substitution; can continue to simplify by making zeroes above diagonal 1 −2 0 1 ????1→ ???? 1 ???? 3 [0 1 −4 4 ] 0 0 1 −1 1 −2 0 1 ????2→ ???? 2 4???? 3 [0 1 0 0 ] 1 0 0 1 1 −1 ????1→ ???? 1 2???? 2 [0 1 0 0| ] 0 0 1 −1 Simplified system shows solut1on: ???? 2 1,???? 3 0,???? = −1 (1,0,−1) o

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