System of Linear Equations Nattee Niparnan. LINEAR EQUATIONS.
1.1 Systems of Linear Equations 1.2 Row Reduction and Echelon Forms Chapter 1: Linear Equations.
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Transcript of 1.1 Systems of Linear Equations 1.2 Row Reduction and Echelon Forms Chapter 1: Linear Equations.
1.1 Systems of Linear Equations1.2 Row Reduction and Echelon
Forms
Chapter 1: Linear Equations
Solve each system of equations
12
5 #1.
yx
yx
363
52 #2.
yx
yx
€
# 3. x + 2y + z = 5
−3x + y − z = 3
x + 3y =1
⎧ ⎨ ⎪
⎩ ⎪
Linear Equation:
System of linear equations: A collection of one or more linear equations involving the same variables
52 :Example xy
bxaxaxa nn 2211
0
5172 :Example
321
31
xxx
xx
numberscomplex or real are ,,, where 1 baa n
•Solution Set The set of all possible solutions to a
linear system.
•Two linear systems are called equivalent, if they have the same solution set.
.equivalent are 02
32&
042
624 :Ex
21
21
21
21
xx
xx
xx
xx
Matrix NotationAny linear system can be written as a
matrix (rectangular array) whose entries are the coefficients.
The size of a matrix tells how many rows and columns it has. (Ex: This is a 2x4 matrix.)
0111
51702
0111
51702
For the linear system:
The matrix
is called thecoefficient matrix, and the matrix
is called the augmented matrix.
111
1702
0
5172
321
31
xxx
xx
Algorithmic Procedure for Solving a Linear System
Basic idea: Replace one system with an equivalent system that is easier to solve.
Example: Solve the following system:
042
624
21
21
xx
xx
042
624
21
21
xx
xx
02
32
21
21
xx
xx
32
02
21
21
xx
xx
33
02
2
21
x
xx
1
02
2
21
x
xx
1
2
2
1
x
x
042
624
021
312
312
021
330
021
110
021
110
201
R1*(1/2) → R1R2*(-1/2) → R2
R1 ↔ R2
-2*R1+R2 → R2
R2*(1/3) → R2
2*R2+R1 → R1
/ 2/(-2)
*(-2)
+)
/3
Elementary Row Operations (Theorem 1.1.1)
360
341
452
360
452
341
360
230
341
360
3/210
3411
2
3
R1 R2 (-2)R1+R2 R2 R2/(-3) R2
Each of the following operations, performed on any linear system, produce a new linear system that is equivalent to the original.
1. Interchange two rows.2. Add a multiple of one row to another row.3. Multiply a row by any nonzero constant.
Gauss-Jordan elimination method to solve a system of linear equations
523
134
4452
Solve
321
321
321
xxx
xxx
xxx
5231
1341
4452Elementary RowOperations
100
010
001
2
2
3
200
200
300
321
321
321
xxx
xxx
xxx
2
2
3
3
2
1
x
x
x
Augmented matrix Reduced row-echelon form
5231
1341
4452
5231
4452
1341
4570
2230
1341
4570
3/23/210
1341
Practice with Elementary Row Operations
5231
2230
1341
5231
2230
1341
3/23/100
3/23/210
1341
2100
2010
3/113/101
Practice with Elementary Row Operations
2100
3/23/210
3/113/101
4570
3/23/210
1341
2100
3/23/210
1341
2100
3/23/210
1341
100
010
001
2
2
3
2100
2010
3/113/101
Practice with Elementary Row Operations
Gauss-Jordan elimination method to solve a system of linear equations
523
134
4452
Solve
321
321
321
xxx
xxx
xxx
5231
1341
4452Elementary RowOperations
100
010
001
2
2
3
200
200
300
321
321
321
xxx
xxx
xxx
2
2
3
3
2
1
x
x
x
Augmented matrix Reduced row-echelon form
Definition. A matrix is in reduced row-echelon form if it has the following four properties:
1. The leftmost nonzero entry in each row is 1. (If the row is not all zeros, this entry is called the leading 1.)
2. Each leading 1 is the only nonzero entry in its column. 3. Each leading 1 of a row is in a column to the right of the
leading 1 of the row above it.4. All nonzero rows are above any rows of all zeros.
Word ProblemsExample. A certain crazed zoologist is raising
experimental animals on a reserve near a toxic waste dump. There are four kinds of animals:
Neon dogs have one head, two tails, four eyes and four legs.
Headless cats have no head, one tail, no eyes and four legs.
Killer ducks have two heads, no tail, four eyes and two legs.
Hopping cows have one head, one tail, two eyes and one leg.
An investigator finds in the inventory book that there are 25 heads, 24 tails, 60 eyes and 72 legs on the reserve. How many of each kind of animal are there?
Existence and Uniqueness
Is the system consistent; that is, does at least one solution exist?
If a solution exists, is it the only one; that is, is the solution unique?
Find the solution set of each of the following:
02
32
21
21
xx
xx
02
32
21
21
xx
xx
624
32
21
21
xx
xx
No solutions
Solution set: {(1, 2)}
Infinitely many solutions.
Consistent
Inconsistent
Unique
Solution Set:
€
x1,x2( ) : x1 =1
2x2 +
3
2,x2 is free
⎧ ⎨ ⎩
⎫ ⎬ ⎭
Solution set: { }
A linear system has one of the following:
No solutions, orExactly one solution, orInfinitely many solutions.
Practice: Writing a solution set
32
1
022
dcb
ca
cba