1 / 15

Lecture #10: System of Linear Equations & Matrices

Lecture #10: System of Linear Equations & Matrices. Lecture #9: Linear Equations: y=mx +b Solution System: N.S., U.S., I.S. Augmented Matrix Solving a System of Linear Equations Today: Echelon Form, Reduced Echelon Form Gauss-Jordan Elimination Method Homogeneous Linear Equations

alize
Download Presentation

Lecture #10: System of Linear Equations & Matrices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture #10: System of Linear Equations & Matrices • Lecture #9: • Linear Equations: y=mx +b • Solution System: N.S., U.S., I.S. • Augmented Matrix • Solving a System of Linear Equations • Today: • Echelon Form, Reduced Echelon Form • Gauss-Jordan Elimination Method • Homogeneous Linear Equations • Matrix Operations

  2. Announcements: • Review Class on Tuesday 28: • Room: Ricketts 203 • Time: 6:30-8:00pm • Exam #1 Next Wednesday: 9/29 Unit Vectors, Cartesian Vector Form.

  3. System of linear equations • The general form: A11x1+A12x2+A13x3+…..A1nxn=B1 Rx=0 A21x1+A22x2+A23x3+…..A2nxn=B2 Ry=0 A31x1+A32x2+A33x3+…..A3nxn=B3 Rz=0 . . . . . . . . . . Am1x1+Am2x2+Am3x3+…Amnxn=Bm

  4. Matrix Form: • Coefficient Matrix ROW # Column #

  5. System of linear eqns. 1x + y + 2z = 9 2x + 4y – 3z = 1 3x + 6y –5z = 0 Remember: Rx=0 Ry=0 Rz=0 Augmented Matrix: (array of numbers of the system of eqns) Augmented Matrix:

  6. Solving a System of Linear Eqns. • GOAL • FIND the solution for x, y,z (TA, TB, TC, TD, TE) • The idea is to replace a given system by a system which has the same solution set, but it is easier to solve.

  7. Basic Operations to find Unknown • Multiply a row by a nonzero constant. (the row you multiply by a number after adding the two rows will not change) • Interchange two rows. • Add a multiple of one row to another row.

  8. Gauss-Jordan Elimination • Goal: to reduce the augmented matrix into a form simple enough such that system of equation are solved by inspection.

  9. Reduced row-echelom form • If row does not consist entirely of zeros, then the first non-zero number in row is 1. • If a row consist of zeros, then they are moved to the bottom of matrix. • In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occur farther to the right of above row. • Each column that contains a leading 1 has zero everywhere.

  10. Reduced Row echelom Must have zeros above and below each leading 1. Row-echelom form Must have zeros below each leading 1. IMPORTANT

  11. Gauss-Jordan Elimination Method • Step1: Locate the leftmost column that does not consist entirely of zero. • Step 2: Interchange the top row with another row, if necessary, to bring a nonzero entry to the top from step 1. • Step 3: If the entry that is now at the top is a constant, divide entire row by it. • Step 4: Add multiples to top row to the rows below such that all entries have 1 as leading term. • Step 5: Cover top row and begin with step 1 applied to submatrix.

  12. Example #1 • For problem 3.22 find FAB, FAC, FAD using Gauss-Jordan method.

  13. Activity:#1 • For Problem in example #1 solve using Gauss-Jordan Method. -TA (0.766) + TB (0.866) = 1699 TA (0.643) + TB (0.500) = 2943

  14. Non-homogeneous A11x1+A12x2+A13x3+…..A1nxn=B1 A21x1+A22x2+A23x3+…..A2nxn=B2 A31x1+A32x2+A33x3+…..A3nxn=B3 . . . . . . . . Am1x1+Am2x2+Am3x3+…Amnxn=Bm The Constants B not equal to 0 Homogeneous A11x1+A12x2+A13x3+…..A1nxn= 0 A21x1+A22x2+A23x3+…..A2nxn= 0 A31x1+A32x2+A33x3+…..A3nxn= 0 . . . . . . . . Am1x1+Am2x2+Am3x3+…Amnxn= 0 The Constants B’s Equal to 0 Homogeneous System of Linear Equations

  15. Trivial Solution X1 = 0 X2 = 0 X3 = 0 …. …. Xn = 0 For same # equations and same # unknowns Non trivial solution: X1 = C1 X2 = C2 X3 = C3 …. …. Xn = C4 When there is more unknowns than equations . Solutions in Homogeneous System

More Related