MAT 2401 Linear Algebra

1. MAT 2401Linear Algebra 1.1, 1.2 Part I Gauss-Jordan Elimination http://myhome.spu.edu/lauw

2. HW • WebAssign 1.2 I • Written Homework

3. Time • Part I may be a bit longer today. • Part II will be shorter next time.

4. Preview • Introduce the Matrix notations. • Study the Elementary Row Operations. • Study the Gauss-Jordan Elimination.

5. Example 1

6. Example 1

7. How many solutions? Q: Given a system of 2 equations in 2 unknowns, how many solutions are possible?

8. How many solutions? Q: Given a system of 3 equations in 3 unknowns, how many solutions are possible?

9. How many solutions? Q: Given a system of 3 equations in 3 unknowns, how many solutions are possible? ______ System ______ System

10. Unique Solution • We will focus only on systems of unique solution in part I. • Such systems appear a lot in applications.

11. Example 2

12. Observation 1 Q: Why eliminations are not good? A: 1. 2. 3.

13. Observation 2 Compare the 2 systems:

14. Observation 2 Compare the 2 systems:

15. Observation 2 Compare the 2 systems:

16. Observation 2 Compare the 2 systems:

17. Extreme Makeover? We want a solution method that • it is systematic, extendable, and easy to automate • it can transform a complicated system into a simple system

18. Extreme Makeover? We want a solution method that • it is systematic, extendable, and easy to automate • it can transform a complicated system into a simple system

19. Extreme Makeover? We want a solution method that • it is systematic, extendable, and easy to automate • it can transform a complicated system into a simple system

20. Extreme Makeover? We want a solution method that • it is systematic, extendable, and easy to automate • it can transform a complicated system into a simple system

21. Gauss-Jordan Elimination

22. Gauss-Jordan Elimination Before we can describe our systematic solution method, we need the matrix notations.

23. Essential Information A system can be represented compactly by a “table” of numbers.

24. Matrix • A matrix is a rectangular array of numbers. • If a matrix has m rows and n columns, then the size of the matrix is said to be mxn.

25. Example 2 Write down the (Augmented) matrix representation of the given system.

26. Coefficient Matrix The left side of the Augmented matrix is called the Coefficient Matrix.

27. Elementary Row Operations We can perform the following operations on the matrix 1. Switching 2 rows. 2. Multiplying a row by a constant. 3. Adding a multiple of one row to another.

28. Elementary Row Operations We can perform the following operations on the matrix 1. Switching 2 rows.

29. Elementary Row Operations We can perform the following operations on the matrix 2. Multiplying a row by a constant.

30. Elementary Row Operations We can perform the following operations on the matrix 3. Adding a multiple of one row to another.

31. Elementary Row Operations Theory: We can use the operations to simplify the system without changing the solution. 1. Switching 2 rows. 2. Multiplying a row by a constant. 3. Adding a multiple of one row to another.

32. Elementary Row Operations Notations (examples) 1. Switching 2 rows. 2. Multiplying a row by a constant. 3. Adding a multiple of one row to another.

33. Gauss-Jordan Elimination

34. Gauss-Jordan Elimination 1 2 3

35. Example 2 3 1 2

36. Remarks • Notice sometimes 2 “parallel” row operations can be done in the same step. • The procedure (algorithm) is designed so that the exact order of creating the “0”s and “1”s is important.

37. Remarks • Try to avoid fractions!!

38. Example 3 Use Gauss-Jordan Elimination to solve the system.

39. Example 3 3 1 2

40. How do I Confirm My Answer?