Linear System of Equations
MGT 4850
Spring 2009
University of Lethbridge
Definition
A linear equation in the variables x1, x2, . . . ,
xn is an equation of the form
a1x1 + a2x2 + ... + anxn = b
where the coefficients a1, a2, . . . , an and
term b of the right-hand side are given
constants.
Example 1.1
x + 2y = 5
4x + y = 6
Example 1.2
x+y+z=4
2x + 2y + 5z = 11
4x + 6y + 8z = 24
Set
A collection of objects, members of the set.
∅ denotes the empty set, i.e., the set with
no members.
a ∈ A means “a is a member of the set A.”
Set Symbols A = B means “the set A is
equal to the set B.”
A ⊆ B means “A is a subset of B.”
A ⊂ B means “A is a proper subset of B.”
Union and intersection
• Let A = {0, 1, 3} and B = {0, 1, 2, 4}. Then
A ∪ ∅ = A,
A ∩ ∅ = ∅,
A ∪ B = {0, 1, 2, 3, 4},
A ∩ B = {0, 1},
A − B = {3}. (take all the members of
set A and subtract the members of A ∩ B)
Natural Numbers
The set of integers
Rational Numbers
Real Numbers
• There is one more problem to overcome.
How do we solve a system like
x2 + 1 = 0
C of complex numbers.
i2 = −1
Homework – define complex numbers
and their use. Provide examples.
Definition
• Any number that can be written in the form
z=a+bi
where a and b are real numbers and i=√-1
The real part is a and the imaginary bi.
Also i2 = −1
i3 = −i
i4 = 1
Euler’s formula z=cos θ +i sin θ (homework)
Quadratic equation
• Solve:
• x 2 -2x+2=0
• x=?
Gaussian Elimination:
4x + 4y = 20
2x − y = 1
• Multiply the first equation by 1/4 to obtain
x+y=5
2x − y = 1
Gaussian Elimination
• Now, multiply a copy of the first equation
x + y = 5 by −2 and add it to the second
eqn 2x − y = 1.
x+y=5
0x − 3y = −9.
This part is called “forward solving.”
Gaussian Elimination
• then work backward
y = −9/−3 = 3.
• Use the first equation to solve for x:
x = 5− y = 5− 3 = 2.
Matrix Notation
Definition
A matrix is a rectangular array of numbers. If
a matrix has m rows and n columns, then
the size of the matrix is said to be m×n. If
the matrix is 1 × n or m × 1, it is called a
vector. If m = n, then it is called a square
matrix of order n. Finally, the number that
occurs in the ith row and jth column is
called the (i, j)th entry of the matrix.
Determinant
• A function that associates a scalar, det(A), to
every nxn square matrix A.
• Characteristic polynomial of the matrix
encodes its eigenvalue, determinant and
trace.
• Homework – define eigenvalue, determinant
and trace. Provide examples.