1. Chapter 1 Linear Algebra S 2 Systems of Linear Equations

2. Ch1_2 Definition ax + by= c ; a,b,c is called a ……………….. The graph of such equation is a ………….……. in xy-plane. The system of two linear equations is like: …………………………………. If ……… satisfy the equations we called them…………... In this system the solution is ………… 1.1 Matrices and Systems of Linear Equations Definition A linear equation in n variables x 1, x 2, x 3, …, x n has the form ………………………………………… where the coefficients a 1, a 2, a 3, …, a n,b

3. Ch1_3 Figure 1.2 ……. solution –2x + y = 3 –4x + 2y = 2 Lines are ……….. No point of intersection. No solutions. Solutions for system of linear equations Figure 1.1 ……… solution x + 3y = 9 –2x + y = –4 Lines ……………. Unique solution: x = 3, y = 2. Figure 1.3 ………. solutions 4x – 2y = 6 6x – 3y = 9 Both equations have the …………………... Any point on the graph is a solution. Many solutions.

4. Ch1_4 The following is an example of a system of three linear equations: How to solve a system of linear equations? For this we introduce a method called ………………………………..

5. Ch1_5 Relations between System of linear equations and Matrices We use matrices to describe system of linear equations: 1. The coefficients of the variables form a matrix called the ……………………….. 2. The coefficients together with the constant terms form a matrix called the ……………………….. Note Ex: matrix of coefficients augmented matrix

6. Ch1_6 Elementary Row Operation 1.……………. two rows of a matrix. 2.………… the elements of a row by a nonzero ……….. 3.Add a ………… of the elements of one row to the corresponding elements of another row. Elementary Row Operations of Matrices Ex:

7. Ch1_7 Example 1 Solve the following system of linear equation by Gauss-Jordan Elimination Solution

8. Ch1_8 Example 2 Solving the following system of linear equation. Solution

9. Ch1_9 Summary This method of solving the system of n linear equations in n variables is called …………………………….. If the system has a ……….solution then A is row equivalent to …………. If A  I n, then the system has ………….. solution. i.e., Def. [I n : X] is called the ……………………….. of [A : B].

10. Ch1_10 Example 3: Many Systems Solving the following three systems of linear equation, all of which have the same matrix of coefficients. Solution

11. Ch1_11 1.2 Gauss-Jordan Elimination Definition A matrix is in reduced echelon form if 1.Any rows consisting entirely of zeros are …………………………. of the matrix. 2.The first nonzero element of each other row is …... This element is called a ………….. 3.The leading 1 of each row after the first is positioned to the……… of the leading 1 of the previous row. 4.All other elements in a column that contains a leading 1 are …….. 5.The reduced echelon form of a matrix is ………..

12. Ch1_12 Examples for reduced echelon form (…..)

13. Ch1_13 Gauss-Jordan Elimination System of linear equations  augmented matrix  reduced echelon form  solution

14. Ch1_14 Example 1 Use the method of Gauss-Jordan elimination to find reduced echelon form of the following matrix. Solution

15. Ch1_15 Example 2 Solve, if possible, the system of equations Solution  ……. sol.

16. Ch1_16 Example 3 Solve, if possible, the system of equations Solution  …… sol.

17. Ch1_17 Homogeneous System of linear Equations Definition A system of linear equations is said to be ……………….... if all the constant terms are ……... Example: Observe that is a solution. Theorem 1.1 A system of homogeneous linear equations in n variables always has the solution x 1 = 0, x 2 = 0. …, x n = 0. This solution is called the …………………...

18. Ch1_18 Homogeneous System of linear Equations Theorem 1.2 A system of homogeneous linear equations that has …………….. than …………… has ……….. solutions. On of these solutions is the trivial solution. Example: