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

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

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

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

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

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

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

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

How to solve any linear system

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

Convert into triangular matrix

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

(-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

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

Echelon Matrix zero row How many zero rows

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

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

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

Echelon Matrix Reduce the augmented matrix to echelon form.

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

Leading variables and Free variables leading Variables Free Variables

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

24 Quiz #1 on Saturday Sec Sec 3.2