chapter 1 linear algebra s 2 systems of linear equations
TRANSCRIPT
Chapter 1
Linear Algebra
S 2
Systems of Linear Equations
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
, , 0a b
Definition A linear equation in n variables x1, x2, x3, …, xn has the form
…………………………………………
where the coefficients a1, a2, a3, …, an ,b
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………. solutions4x – 2y = 66x – 3y = 9 Both equations have the …………………... Any point on the graph is a solution. Many solutions.
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 ………………………………..
62
3 32
2
321
321
321
xxx
xxx
xxx
Ch1_5
1 2 3
1 2 3
1 2 3
2
2 3 3
2 6
x x x
x x x
x x x
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
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
ij i jR R R
ikR
i jkR R
Ex: 1 1 1
2 3 1
1 1 2
3 21 2 2 22
................... ...................... ......................R RR R R
Ch1_7
Example 1Solve the following system of linear equation by Gauss-Jordan Elimination
623322
321
321
321
xxxxxxxxx
Solution
Ch1_8
Example 2Solving the following system of linear equation.
83318521242
321
321
321
xxxxxxxxx
Solution
1
2
3
2
solution 1
3
x
x
x
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 In, then the system has ………….. solution.
[ : ] [...... : .......]A B i.e.,
Def. [In : X] is called the ……………………….. of [A : B].
Ch1_10
Example 3: Many SystemsSolving the following three systems of linear equation, all of which have the same matrix of coefficients.
3321
2321
1321
42
for 42
3
bxxx
bxxx
bxxx
in turn 433
,210
,11118
3
2
1
bbb
Solution
Ch1_11
1.2 Gauss-Jordan EliminationDefinitionA 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 ………..
Ch1_12
Examples for reduced echelon form
10000
04300
03021
9100
3010
7001
3100
0000
4021
000
210
801
(…..) (…..)(…..) (…..)
Ch1_13
Gauss-Jordan Elimination
System of linear equations augmented matrix reduced echelon form solution
Ch1_14
Example 1Use the method of Gauss-Jordan elimination to find reduced echelon form of the following matrix.
1211244129333
22200
Solution
Ch1_15
Example 2Solve, if possible, the system of equations
7537429333
321
321
321
xxxxxxxxx
Solution
……. sol.
Ch1_16
Example 3Solve, if possible, the system of equations
1 2 3
2 3
1 2 3
5 3
3 1
2 8 3
x x x
x x
x x x
Solution
…… sol.
Ch1_17
Homogeneous System of linear Equations
Definition A system of linear equations is said to be ……………….... if
all the constant terms are ……...
Example:
0632
052
321
321
xxx
xxx
Observe that is a solution. 1 2 3....., ....., .....x x x
Theorem 1.1
A system of homogeneous linear equations in n variables always has the solution x1 = 0, x2 = 0. …, xn = 0. This solution is called the …………………...
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.
0632
052
321
321
xxx
xxxExample: