Date post: | 11-Jan-2016 |
Category: |
Documents |
Upload: | gyles-cunningham |
View: | 227 times |
Download: | 0 times |
GUC - Spring 2012 3
Linear System of Equations
• A system of m-linear equations in n variables x1, x2, ..., xn has the general form
(1)
GUC - Spring 2012 4
Linear System of Equations
• where the coefficients aij(i = 1, 2, ...,m; j = 1, 2, ..., n)
• and the quantities bi
• are all known scalars (numbers).
GUC - Spring 2012 7
Matrix Representation of a Linear System of Equations
• Any linear system of the form (1) can be written in the matrix form
AX = B
GUC - Spring 2012 8
Matrix Representation of a Linear System of Equations
• With
A is the coefficient matrix
GUC - Spring 2012 9
Matrix Representation of a Linear System of Equations
X is the column of variables
GUC - Spring 2012 10
Matrix Representation of a Linear System of Equations
B is the column of constants
GUC - Spring 2012 13
Gauss Elimination Method
• The method consists of four steps 1. Construct an augmented matrix for the
given system of equations. 2. Use elementary row operations to
transform the augmented matrix into an augmented matrix in row-reduced form.
3. Write the equations associated with the resulting augmented matrix.
4. Solve the new set of equations by back substitution.
GUC - Spring 2012 14
Augmented Matrix
• The augmented matrix for AX = B is the partitioned matrix [A|B]
• E.g.
• has its augmented matrix as
GUC - Spring 2012 15
Elementary Row Operationselementary row operations are :
• 1- Interchange any two rows in a matrix
• 2- Multiply any row of a matrix by a nonzero scalar
• 3- Add to one row of a matrix a scalar times another row of the same matrix
GUC - Spring 2012 16
Row-Reduced Form• A matrix is in row-reduced form if it satisfies the
following four conditions:
1. All zero rows appear below nonzero rows when both types are present in the matrix.
2. The first nonzero element in any nonzero row is 1.
3. All elements directly below (that is, in the same column but in succeeding rows form) the first (left- to- right)
nonzero element of a nonzero row are 0 .
4. The first nonzero element of any nonzero row appears in a later column (further to right) than the first nonzero
element in any preceding row.
GUC - Spring 2012 19
Step 2Elementary row operations
We use elementary row operations to transform the augmented matrix into
row-reduced form as follows,
GUC - Spring 2012 26
Step 3The Resulting System• We write the equations of the resulting
augmented matrix
GUC - Spring 2012 27
Step 3Writing the Resulting System
• We write the equations associated with the resulting augmented matrix
GUC - Spring 2012 28
Step 4Solving the Resulting System
• we Solve the derived set of equations by back substitution.
• The third equation implies that z = 5,
• Substituting in the second equation, we get y = 12 − 15 = −3,
• substituting with the values of z and y in the first equation, we get x = 4
GUC - Spring 2012 29GUC - Spring 2012 29
Example
• Use Gauss elimination method to solve the linear system of equations
GUC - Spring 2012 34GUC - Spring 2012 34
Writing the Resulting System
• The resulting system of equations is
GUC - Spring 2012 35GUC - Spring 2012 35
Solving the Equations
• Solving the last system by back substitution, we get the solution
• x = 1 and y = 1
GUC - Spring 2012 36GUC - Spring 2012 36
Example
• Use Gauss elimination method to solve the linear system of equations
GUC - Spring 2012 41GUC - Spring 2012 41
Important Note
• Since the final system has the number of variables (4) greater than the number of equations (3),
• then one of the variable will be arbitrary,
• and the other variables will be found in terms of it.
GUC - Spring 2012 42GUC - Spring 2012 42
Solving The Resulting System
• The solution will be found in terms of x4 as follows,
GUC - Spring 2012 43
Note
• Since x4 is arbitrary, then the system has infinite number of solutions, depending on the value of x4
• for example if you choose
• Then,
GUC - Spring 2012 43