1. Arab Open University Faculty of Computer Studies M132: Linear Algebra

2. Course Contents Systems of Linear Equations
Matrices and Matrix Operations
Vectors, Linear Combinations and Linear Independence
Vector Spaces, Subspaces, Span, Basis and Dimensions
Linear Transformations, Null spaces and Ranges
Eigenvalues and Eigenvectors

3. Systems of Linear Equations
A set of equations is called a system of equations.
A linear equation in n unknowns has the form
where the variables are of first-degree.
If all equations in a system are linear, the system is a system of linear equations, or a linear system.
The solutions must satisfy each equation in the system.

4. Systems of Linear Equations
Example Solve the linear system
Solution
(1)
(2)
Solve (2) for y
Substitute y = x + 3 in (1)
Solve for x
Substitute x = 1 in y = x + 3
Solution set: {(1, 4)}

5. Systems of Linear Equations
Matrices
Information in science and mathematics is often organized into rows and columns to form rectangular arrays.
Tables of numerical data that arise from physical observations
2×3

6. Linear Systems ~ Matrices
Systems of Linear Equations
Linear Systems ~ Matrices
Example: (to solve linear equations)
Solution is obtained by performing appropriate operations on this matrix

7. Systems of Linear Equations
For any system of linear equations, we have 3 possibilities
Unique solution
Infinitely many solutions
No solution y y y
(2)
(1)
(2)
(1)
(1)
(2) x x x
(1,-1)
x = 1, y = −1
Let y = t, x = 3 − t, t in R
System is inconsistent

8. Systems of Linear Equations

9. Systems of Linear Equations

10. Systems of Linear Equations

11. Systems of Linear Equations

12. Systems of Linear Equations
Definition: An m×n matrix A is said to be in reduced row echelon form if it satisfies the following conditions:
All zero rows (consisting entirely of zeros), if any, are at the bottom.
The first nonzero entry from the left of a nonzero row is 1, called the leading 1 for that row.
Each leading 1 is to the right of all leading 1’s in the rows above it.
Each leading 1 is to the only nonzero entry in its column.
e.g.

13. Systems of Linear Equations

14. Systems of Linear Equations
Definition: The elementary row operations on an m×n matrix A are:
Interchanging two rows.
Multiplying one row by a nonzero number.
Add a multiple of one row to a different row.
The matrix B is row equivalent to the matrices A.

15. Systems of Linear Equations
Elementary Row Operations (Example)
R2 = R2 – 2R1
R3 = R3 – 3R1

16. Systems of Linear Equations

17. Systems of Linear Equations

18. Systems of Linear Equations

19. Systems of Linear Equations

20. Systems of Linear Equations
Let AX = B and CX = D be two systems of linear equations each of m equations in n unknowns. If the augmented matrices [A | B ] and [C | D ] of these systems are row equivalent, then both linear systems have exactly the same solutions.
To solve the system AX = B :
Form the augmented matrix [A | B ].
Find the matrix [C | D ] in reduced row echelon form that is row equivalent to the matrix [A | B ] that by using elementary row operations.
For the matrix [C | D ], there are 3 possibilities:
Number of leading 1’s = number of unknowns (variables), then the system has the unique solution X = D.
Number of leading 1’s < number of unknowns, then the system has infinitely many solutions. Here the non-leading variables (unknowns corresponding to columns that do not contain leading 1) end up as parameters and the leading variables (unknowns corresponding to columns that contain leading 1) are given in terms of these parameters.
The system is inconsistent (0 = 1 !!!), the system has no solution.

21. Systems of Linear Equations
AX = B [A | B ] [C | D ]
Unique solution
Infinitely many solutions
No solution

22. Systems of Linear Equations

23. Systems of Linear Equations

24. Systems of Linear Equations

25. Systems of Linear Equations

26. Systems of Linear Equations

27. Systems of Linear Equations

28. Systems of Linear Equations
For the system of linear equations AX = B (B ≠ O),
If X1 and X2 are two solutions, then rX1 + sX2 , r + s = 1, is also a solution.
e.g. If X1 and X2 are two solutions to the system of linear equations AX = B (B ≠ O), then 3X1 - 2X2 and 0.25X X2 are also solutions:
e.g. If X1, X2 and X3 are solutions to the system of linear equations AX = B (B ≠ O), then 3X1 + 2X2 - 4X3 is also a solution.

29. Systems of Linear Equations
For the homogenous system of linear equations AX = O,
If X1 and X2 are two solutions, then rX1 + sX2 is also a solution.
e.g. If X1 and X2 are two solutions, then 3X1 + 2X2 and 10X1 - 5X2 are also solutions.
The homogenous system is always consistent (has solution) which is either of following: The unique solution ( X = O, Zero solution), called the trivial solution, or an infinitely many solutions (including the trivial solution), called the nontrivial solution.

30. Systems of Linear Equations

31. Systems of Linear Equations

32. Systems of Linear Equations

33. Systems of Linear Equations