1. Sec 3.1 Introduction to Linear System Sec 3.2 Matrices and Gaussian Elemination The graph is a line in xy-plane The graph is a line in xyz-plane

2. Sec 3.1 Introduction to Linear System Coefficient Matrix 3 x 3 Coefficient Matrix 3 x 3 Augmented Coefficient Matrix 3 x 4 Augmented Coefficient Matrix 3 x 4 Sec 3.2 Matrices and Gaussian Elemination

3. Sec 3.1 Introduction to Linear System Augmented Coefficient Matrix 3 x 4 Augmented Coefficient Matrix 3 x 4 Sec 3.2 Matrices and Gaussian Elemination Size, shape row column

4. Coefficient Matrix n x n Coefficient Matrix n x n Augmented Coefficient Matrix n x (n+1) Augmented Coefficient Matrix n x (n+1) Sec 3.1 Introduction to Linear System Sec 3.2 Matrices and Gaussian Elemination

5. Sec 3.1 Introduction to Linear System Sec 3.2 Matrices and Gaussian Elemination

6. Three Possibilities Linear System Unique Solution 1 Infinitely many solutions 2 No Solution 3 Inconsistent consistent

7. How to solve any linear system Triangular system Use back substitution Augmented

8. Elementary Row Operations Multiply one row by a nonzero constant 1 1 Interchange two rows 2 2 Add a constant multiple of one row to another row 3 3 Triangular system

9. How to solve any linear system

10. (-3) R1 + R2 (-2) R1 + R3 (-3) R2 + R3 Augmented Matrix (1/2) R2 Convert into triangular matrix triangular matrix

11. Convert into triangular matrix

12. How to solve any linear system Triangular system Use back substitution Augmented Solve

13. (-3) R1 + R2 (-2) R1 + R3 (-3) R2 + R3 Augmented Matrix (1/2) R2 Definition: (Row-Equivalent Matrices) A and B are row equivalent if B can be obtained from A by a finite sequence of elementary row operations A B Convert into triangular matrix A and B are row equivalent

14. Definition: (Row-Equivalent Matrices) A and B are row equivalent if B can be obtained from A by a finite sequence of elementary row operations A B A and B are row equivalent A is the augmented matrix of sys(1) B is the augmented matrix of sys(2) Theorem 1: A and B are row equivalent & sys(1) and sys(2) have same solution

15. Echelon Matrix zero row How many zero rows

16. Echelon Matrix non-zero row 1)How many non-zero rows 2)Find all leading entries leading entry The first (from left) nonzero element in each nonzero row

17. Echelon Matrix Def: A matrix A in row-echelon form if 1)All zero rows are at the bottom of the matrix 2)In consecutive nonzero rows the leading in the lower row appears to the right of the leading in the higher row

18. How to transform a matrix into echelon form Gaussian Elimination 1) Locate the first nonzero column 2) In this column, make the top entry nonzero 3) Use this nonzero entry to (below  zeros ) 4) Repeat (1-3) for the lower right matrix

19. Echelon Matrix Reduce the augmented matrix to echelon form.

20. How to solve any linear system Gaussian Elimination Use back substitution Augmented

21. Leading variables and Free variables leading Variables Free Variables

22. Back Substitution 1) Set each free variable to parameter ( s, t, …) 2) Solve for the leading variables. Start from last row. Second row gives: first row gives: Thus the system has an infinite solution set consisting of all (x,y,z) given in terms of the parameter s as follows

23. 23 Back Substitution The linear systems are in echelon form, solve each by back substitution

24. 24 Quiz #1 on Saturday Sec 3.1 + Sec 3.2