1. Math 201 for Management Students
Integration & Linear Algebra & Statistics
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2. Lecture 6 Linear Algebra (III)
Linear Systems of Equations
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3. Linear System of Equations
A system of m-linear equations in n variables x1, x2, ..., xn has the general form
(1)
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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).
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5. This is a Linear System
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6. This is not a Linear System
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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
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8. Matrix Representation of a Linear System of Equations
With
A is the coefficient matrix
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9. Matrix Representation of a Linear System of Equations
X is the column of variables
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10. Matrix Representation of a Linear System of Equations
B is the column of constants
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11. Example
Can be represented as
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12. Example
The matrix form
Represents the system
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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.
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14. Augmented Matrix
The augmented matrix for AX = B is the partitioned matrix [A|B]
E.g.
has its augmented matrix as
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15. Elementary Row Operations
elementary 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
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16. Row-Reduced Form
A matrix is in row-reduced form if it satisfies the following four conditions:
All zero rows appear below nonzero rows when both types are present in the matrix.
The first nonzero element in any nonzero row is 1.
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 .
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.
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17. Example
Use Gaussian elimination to solve the system
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18. Step 1 The augmented matrix
The augmented matrix of the system is:
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19. Step 2 Elementary row operations
We use elementary row operations to transform the augmented matrix into row-reduced form as follows,
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25. This is a Row Reduced Form
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26. Step 3 The Resulting System
We write the equations of the resulting augmented matrix
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27. Step 3 Writing the Resulting System
We write the equations associated with the resulting augmented matrix
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28. Step 4 Solving 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
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29. Example
Use Gauss elimination method to solve the linear system of equations
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30. Example
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31. The Augmented Matrix
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32. Getting Row Reduced Form
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33. Getting Row Reduced Form
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34. Writing the Resulting System
The resulting system of equations is
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35. Solving the Equations
Solving the last system by back substitution, we get the solution
x = 1 and y = 1
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36. Example
Use Gauss elimination method to solve the linear system of equations
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37. The Augmented Matrix
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38. Getting Row Reduced Form
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39. Getting Row Reduced Form
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40. Writing the Resulting System
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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.
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42. Solving The Resulting System
The solution will be found in terms of x4 as follows,
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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,
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44. Japan !
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45. No Comment
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