Download Section 1.1: Systems of Linear Equations Two Linear Equations in

Section 1.1: Systems of Linear Equations
Two Linear Equations in Two Unknowns
Recall that the equation of a line in 2D can be written in standard form: a1 x1 + a2 x2 = b.
Definition. A 2 × 2 system of linear equations has two equations and two unknowns (x1 , x2 ):
a11 x1 + a12 x2 = b1
a21 x1 + a22 x2 = b2
The aij ’s and bi ’s are real numbers and the xi ’s are variables. A solution to a 2 × 2 linear
system is a 2-tuple (x1 , x2 ) of numbers that satisfies both equations in the system. The set of all
solutions is called the solution set. A system is called consistent if it has at least one solution,
or inconsistent otherwise.
Three situations can occur: the two equations have lines (a) intersecting at a single point, (b)
parallel and intersecting nowhere, or (c) parallel and are the same line.
(a)
(b)
(c)
x1 + x2 = 5
x1 + x2 = 5
x1 + x2 = 5
x1 − x2 = 5
x1 + x2 = 0
−x1 − x2 = −5
consistent
solution(s): (5, 0)
inconsistent
solution(s): none
consistent
solution(s): {(α, 5 − α) : α ∈ R}
In general, these are the only three situations that can occur. If the system is consistent, it has
either exactly one solution or an infinite number of solutions.
M304 Notes
©, R.G. Lynch, Texas A&M
Section 1.1: Systems of Linear Equations Page 2 of 5
General Linear Systems
Nothing is special about the case of two equations with two unknowns. We can generalize all of
these concepts to m equations with n unknowns.
Definition. A linear equation with n unknowns (that is, in n dimensions) is of the form
a1 x1 + a2 x2 + · · · + an xn = b
where ai , b ∈ R are numbers are the xi ’s are variables. A linear system with m equations in n
unknowns is of the form
a11 x1 + a12 x2 + · · · + a1n xn = b1
a21 x1 + a22 x2 + · · · + a2n xn = b2
..
.
am1 x1 + am2 x2 + · · · + amn xn = bm
A solution of the system is an n-tuple (x1 , x2 , . . . , xn ) that satisfies all m equations. Consistency
is defined in the same way as 2 × 2 systems and again, we can only have exactly one solution, an
infinite number of solutions, or no solutions.
Example. Let’s examine the solution sets of the following linear systems
(a)
3x1 + 2x2 − x3 = −2
x2
=3
2x3 = 4
(b)
3x1 + 2x2 − x3 = −2
−3x1 − x2 + x3 = 5
3x1 + 2x2 + x3 = 2
M304 Notes
©, R.G. Lynch, Texas A&M
Section 1.1: Systems of Linear Equations Page 3 of 5
Definition. Two systems of equations involving the same variables are equivalent if they have
the same solution set.
There are three operations that can be used on a system to obtain an equivalent system:
I. The order of the equations can be changed.
II. Both sides of an equation may be multiplied by the same nonzero real number.
III. A multiple of one equation can be added/subtracted to another.
n × n Systems
If an n × n system has exactly one solution, we can rewrite it to obtain an equivalent “strictly
triangular system.”
Definition. A system is said to be in strict triangular form if, in the kth equation, the
coefficients of the first k − 1 variables are all zero and the coefficient of xk is nonzero.
If an n × n linear system can be reduced to to this form, then it will have a unique solution.
Example. Solve the following systems. We use a method called back substitution.
(a)
3x1 + 2x2 + x3 = 1
x2 − x3 = 2
2x3 = 4
(b)
2x1 − x2 + 3x3 − 2x4
x2 − 2x3 + 3x4
4x3 + 3x4
4x4
=1
=2
=3
=4
M304 Notes
(c)
©, R.G. Lynch, Texas A&M
Section 1.1: Systems of Linear Equations Page 4 of 5
x1 + 2x2 + x3 = 3
3x1 − x2 − 3x3 = −1
2x1 + 3x2 + x3 = 4
Definition. Looking at the last example, we can associate with it a 3 × 3 array of numbers who
entries are the coefficients of the xi ’s:


1
2
1
3 −1 −3
2
3
1
called the coefficient matrix of the system. A matrix is simply a rectangular array of numbers.
A matrix with m rows and n columns is said to be m × n and if m = n it is called square.
Definition. We can also attach to the coefficient matrix the another column with the bi ’s in it:


1
2
1
3
3 −1 −3 −1
2
3
1
4
This is called an augmented matrix. We can actually attach any two matrices with the same
number of rows in this way, not just a square one and another column. That is, if A is an m × r
matrix and B is an m × s matrix, then the augmented matrix (A | B) is a m × (r + s) matrix.
Definition. Since the augmented matrix of a linear system is just a convenient way to represent
a linear system of equations, we can apply the same three operations to it. These are called
Elementary Row Operations:
I. Interchange two rows.
II. Multiple a row by a nonzero real number.
III. Replace a row by its sum with a multiple of another row.
We will typically use these to get the augmented matrix into strictly triangular form and then use
back substitution.
M304 Notes
©, R.G. Lynch, Texas A&M
Section 1.1: Systems of Linear Equations Page 5 of 5
Example. Write the augmented matrix of the system and use it to solve.
− x2 − x3 + x4
x1 + x2 + x3 + x4
2x1 + 4x2 + x3 − 2x4
3x1 + x2 − 2x3 + 2x4
=0
=6
= −1
=3