Revision
Lecture 01
Revision
Lecture 01
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Review
ˆ Gauss-Jordan Elimination
ˆ Row equivalence
ˆ Echelon form
ˆ Rank Theorem
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System of Linear Equations
Definition
A linear equation in n variables x1 , x2 , . . . , xn is an equation that can be
written in the form
a1 x1 + a2 x2 + · · · + an xn = b
where the coefficients a1 , a2 , . . . , an and the constant term b are
constants.
Definition
A
of a linear equation a1 x1 + a2 x2 + · · · + an xn = b is a vector
 solution

s1
 s2 
 
 ·  whose components satisfy the equation when we substitute x1 = s1 ,
 
· 
sn
x2 = s2 , · · · , xn = sn .
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System of Linear Equations
Definition
A system of linear equations is a finite set of linear equations, each with
the same variables.
A solution of a system of linear equations is a vector that is
simultaneously a solution of each equation in the system.
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System of Linear Equations
Example 2.6
Solve the system
x
3x
2x
−y
−3y
−y
−z
+2z
+z
=2
= 16
=2
Solution
The augmented matrix of the system is




1 −1 −1 2
1 −1 −1 2
R −3R1
R −2R1
 3 −3
0
2 16  −−2−−−→
0
5 10  −−3−−−→
2 −1
1 9
2 −1
1 9




1 −1 −1 2
1 −1 −1 2
R2 ↔R3
0
0
5 10  −−
1
3 9
−−→  0
0
1
3 9
0
0
5 10
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System of Linear Equations
Solution
The matrix is in row echelom form.
Using back substitution we have
x
−y
y
−z
+3z
5z
=2
=5
= 10
z = 2, y = −1, x = 3.


3
Therefore x =  −1 .
2
Definition(Row equivalence)
Matrices A and B are row equivalent if there is a sequence of
elementary row operations that converts A into B.
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System of Linear Equations
Elementary row operations
(a) Ri ←→ Rj , i ̸= j.
(b) kRi , k ̸= 0.
(c) Ri + kRj , i ̸= j.
Examples
of row equivalent
matrices
2 4
2 0
1 3
0 0
A=
,B=
,D=
.
,C=
0 1
0 1
0 1
0 21
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System of Linear Equations
Theorem 2.1
Matrices A and B are row equivalent if and only if they can be reduced to
the same row echelon form.
Definition
A matrix is in row echelon form if it satisfies the following properties
(1) Any rows consisting entirely of zeros are at the bottom.
(2) In each non-zero row, the first non-zero entry (leading entry) is a
column to the left of any leading entries below it.
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System of Linear Equations
Example

 

1 1 2 1
1 0 1
 0 0 1 3 ,  0 1 5 .
0 0 0 0
0 0 4
Example 2.11
Solve the system
w
2w
−w
−x
−2x
+x
−y
−y
−y
Revision
+2z
+3z
=1
=3
= −3
Lecture 01
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System of Linear Equations
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System of Linear Equations
w
−x
−y
y
Revision
+2z
−z
=1
=1
Lecture 01
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Elementary row operations
Example
2 2
2 1
2 0
Let A =
,B=
,C=
.
0 1
0 12
0 12
2 0 R1 +2R2 2 2 12 R2 2 2
A=
−−−−−→
−−→
=B
0 1
0 1
0 21
2 0 12 R2 2 0 R1 +2R2 2 1
−−→
A=
=C
−−−−−→
0 1
0 21
0 21
This shows that elementary row operations that do not commute.
Row echelon form of a matrix is not unique.
Example
1 0
2 0
,
.
0 3
0 1
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Rank of a matrix
Definition
The rank of a matrix A, denoted by rank(A), is the number of non-zero
rows in its row echelon form.
Example

1 1
A = 0 0
0 0
rank(A) = 3



2 1
1 0 1
1 3 , B =  0 1 5 .
0 1
0 1 5
and rank(B) = 2
Theorem 2.2(The Rank Theorem)
Let A be the coefficient matrix of a system of linear equations with n
variables. If the system is consistent, then
number of free variables = n − rank(A).
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Rank of a matrix
Example
Solve the system
x
−y
3y
2y
+2z
−3z
−2z
=3
= −6
=1
Solution
The augmented matrix of the system is




1 −1
2
3
1 −1
2
3
0
3 −3 −6  −−−−−→  0
3 −3 −6 
R3 − 32 R2
0
2 −2
0
0
0
1
5
The system has no solutions. Why?
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Lecture 01
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