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Unit 2 Linear Equations
Jan, 2011.
Linear Equation in n variables
• a1x1 + a2x2+ … + an xn = c
– a1, a2, …, an are called coefficients (real numbers).
– x1, x2,…, xn are variables (or indeterminates).
– c is a constant term (real number).
• Example
– 2x + 3y – 4z = 0.2
• Non-example
– x2+y2=1
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Geometry of a linear equation
Three variables: plane
Two variables: straight line
ax + by = c
ax + by + cz = d
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System of linear equations
• A system of linear equations (or linear system)
is a collection of one or more linear equations.
– for example:
• A solution is a list of numbers
(s1, s2, …, sn) which satisfies all equalities after
substituting xi by si, for i =1,2,…,n.
• The set of all solutions is called the solution
set.
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Nutrition problem
• Find a combination of food A, B, C and D in order to satisfy the
nutrition requirement exactly.
Food A
Food B
Food C
Food D
Requirement
Protein
9
8
3
3
5
Carbohydrate
15
11
1
4
5
Vitamin A
0.02
0.003
0.01
0.006
0.01
Vitamin C
0.01
0.01
0.005
0.05
0.01
• Let xA, xB, xC and xD be the amount of food A, B, C and D
respectively.
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Formal notation
• Given a system of m linear equations in n
variables
Double
subscripts
the solution set is defined as
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Review of set notation
• Set of Greek letters = {,,,,,,,,,,
,,,,,,,,,,,,,}
finite
• Set of prime numbers =
{2,3,5,7,11,13,17,23,29,31,37,41, …}
Countably infinite
• Sphere with radius r centered at origin
= {(x,y,z): x2+y2+z2=r2}
Uncountably infinite
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Examples of solution sets
{ (x,y): ax + by = c }
z
5
0
y
-5
1
0.5
1
0.5
0
0
-0.5
y
-0.5
-1
-1
x
x
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Consistency
• A linear system is called consistent if there is
at least one solution, in other words, if the
solution set is non-empty.
y
y
Inconsistent,
no solution
Consistent
x
x
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Classification
Linear
System
Inconsistent
Consistent
(no solution)
Tasks:
Determine whether a linear
system is consistent.
If yes, find all solutions.
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Unique solution
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Infinitely many
solutions
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Short-hand notation using matrix
(2 rows, 4 columns)
Usually called the augmented matrix
(4 rows, 3 columns)
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The nutrition example
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Elementary row operations
1. Interchange two rows
2. Multiply a row by a non-zero constant
3. Replace a row by the sum of itself and a
constant multiple of another row
Facts: Elementary row operations do not change the solution(s).
(There is no loss, and no gain, of information.)
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Illustration – row interchange
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Illustration – Multiply by constant
2
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2
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Illustration – Row replacement
(1)  (1) – (2)
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(1)  (1) – (2)
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How to solve?
• Idea: Apply the three kinds of informationlossless elementary row operations, and
transform the linear system into one which is
easier to solve.
Linear system in upper triangular
matrix form can be easily solved by
backward substitution
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Carl Friedrich Gauss
(1777~1855)
The old Deutsche 10-Mark note
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Gaussian elimination
• Step 0: Write the linear system in matrix
format
• Step 1: Try to transform the matrix into upper
triangular form
• Step 2: Solve for the variables one by one, in
backward order
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Example
1
(row
operations)
Solve
(1)
(2)
(3)
(2)  (2) – (1)
(3)  (3) + (2)/2
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Example 1 (backward sub.)
Upper triangular
Solution: x=1/3, y = –5/3, z = 7/3
(unique solution)
(3)  z = 7/3
(2)  – 2y – (7/3) = 1  y = –5/3
(1)  x+(–5/3)+(7/3) = 1  x = 1/3
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Verify:
x+y+z = 1/3 – 5/3 + 7/3 = 3/3 = 1
x–y = (1/3) – (– 5/3) = 6/3 = 2
y+2z = (– 5/3)+2(7/3) = 9/3 = 3
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Example 2 (row operations)
Solve
(2)  (2) – (1)
(3)  (3) – (1)
(3)  (3) – 2  (2)
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Example 2 (backward sub.)
(1)
(2)
z can be taken as a
free variable.
Let z to be any real
number.
Solution:
x= 2– 2z,
y = –1–z,
z = any real number.
From (2), y = –1 – z
Solution set = {(2 – 2z, –1–z, z): z is any real no.}
(Infinitely many solutions)
From (1), x +(–1 – z)+3z = 1  x = 2 – 2z
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Note: You can let y to
be the free variable as well,
and obtain the solutions in
terms of y.
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Example 2 (cont’d)
Solution: x= 2– 2z, y = –1–z, z = any real number
Solution set = {(2 – 2z, –1–z, z): z is any real no.}
• Verification
4
2
0
-2
z
– x+y+3z
=(2 – 2z) + (– 1 – z) + 3z
=1
– x+2y+4z
=(2 – 2z) + 2(– 1 – z) + 3z
=0
– x+3y+5z
=(2 – 2z) + 3(– 1 – z) + 5z
=–1
-4
-6
-8
-5
-10
-5
0
0
5
5
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y
10
x
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Example 3 (row operations)
Solve
(2)  (2) – (1)
(3)  (3) – (1)
(3)  (3) – 2(2)
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Example 3 (cont’d)
Contradiction, cannot be true
Answer:
the linear system is inconsistent
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Example 3 (picture)
2.5
2
1.5
z
Cross-section
1
0.5
0
No common intersection
-1
-1.2
-1.4
1
-1.6
0
-1
-2
-1.8
-3
y
-2
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An infinitely long triangular tube
is formed by the three planes
-4
-5
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Key concepts
• Three kinds of elementary row operations
– The solution set is invariant under any elementary row
operation
• Gaussian elimination
– Transform a linear system to upper triangular form
– Backward substitution
• Three types of solutions
– No solution
– Unique solution
– Infinitely many solutions
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