This section and the next are devoted to the study of systems of linear equations, which arise naturally in both the physical and social sciences. In this section, we apply results from Chapter 2 to describe the solution sets of systems of linear equations as subsets of a vector space. In Section 3.4, we will use elementary row operations to provide a computational method for finding all solutions to such systems.
The system of equations
where aij
The m×n
is called the coefficient matrix of the system (S).
If we let
then the system (S) may be rewritten as a single matrix equation
To exploit the results that we have developed, we often consider a system of linear equations as a single matrix equation.
A solution to the system (S) is an n-tuple
such that As=b
(a) Consider the system
By use of familiar techniques, we can solve the preceding system and conclude that there is only one solution: x1=2, x2=1
In matrix form, the system can be written
so
(b) Consider
that is,
This system has many solutions, such as
(c) Consider
that is,
It is evident that this system has no solutions. Thus we see that a system of linear equations can have one, many, or no solutions.
We must be able to recognize when a system has a solution and then be able to describe all its solutions. This section and the next are devoted to this end.
We begin our study of systems of linear equations by examining the class of homogeneous systems of linear equations. Our first result (Theorem 3.8) shows that the set of solutions to a homogeneous system of m linear equations in n unknowns forms a subspace of Fn
A system Ax=b
Any homogeneous system has at least one solution, namely, the zero vector. The next result gives further information about the set of solutions to a homogeneous system.
Let Ax=0
Clearly, K={s∈Fn: As=0}=N(LA)
If m<n
Suppose that m<n
where K=N(LA)
(a) Consider the system
Let
be the coefficient matrix of this system. It is clear that rank(A)=2
is a solution to the given system,
is a basis for K. Thus any vector in K is of the form
where t∈R
(b) Consider the system x1−2x2+x3=0
are linearly independent vectors in K. Thus they constitute a basis for K, so that
In Section 3.4, explicit computational methods for finding a basis for the solution set of a homogeneous system are discussed.
We now turn to the study of nonhomogeneous systems. Our next result shows that the solution set of a nonhomogeneous system Ax=b
Let K be the solution set of a consistent system of linear equations Ax=b
Let s be any solution to Ax=b
So w−s∈KH
Conversely, suppose that w∈{s}+KH;
(a) Consider the system
The corresponding homogeneous system is the system in Example 2(a). It is easily verified that
is a solution to the preceding nonhomogeneous system. So the solution set of the system is
by Theorem 3.9.
(b) Consider the system x1−2x2+x3=4
is a solution to the given system, the solution set K can be written as
The following theorem provides us with a means of computing solutions to certain systems of linear equations.
Let Ax=b
Suppose that A is invertible. Substituting A−1b
Conversely, suppose that the system has exactly one solution s. Let KH
Consider the following system of three linear equations in three unknowns:
In Example 5 of Section 3.2, we computed the inverse of the coefficient matrix A of this system. Thus the system has exactly one solution, namely,
We use this technique for solving systems of linear equations having in-vertible coefficient matrices in the application that concludes this section.
In Example 1(c), we saw a system of linear equations that has no solutions. We now establish a criterion for determining when a system has solutions. This criterion involves the rank of the coefficient matrix of the system Ax=b
Let Ax=b
To say that Ax=b
the span of the columns of A. Thus Ax=b
So by Theorem 3.5, the preceding equation reduces to
Recall the system of equations
in Example 1(c).
Since
rank(A)=1
We can use Theorem 3.11 to determine whether (3, 3, 2) is in the range of the linear transformation T: R3→R3
Now (3, 3, 2)∈R(T)
Since the ranks of the coefficient matrix and the augmented matrix of this system are 2 and 3, respectively, it follows that this system has no solutions. Hence (3, 3, 2)∉R(T)
In 1973, Wassily Leontief won the Nobel prize in economics for his work in developing a mathematical model that can be used to describe various economic phenomena. We close this section by applying some of the ideas we have studied to illustrate two special cases of his work.
We begin by considering a simple society composed of three people (industries)—a farmer who grows all the food, a tailor who makes all the clothing, and a carpenter who builds all the housing. We assume that each person sells to and buys from a central pool and that everything produced is consumed. Since no commodities either enter or leave the system, this case is referred to as the closed model.
Each of these three individuals consumes all three of the commodities produced in the society. Suppose that the proportion of each of the commodities consumed by each person is given in the following table. Notice that each of the columns of the table must sum to 1.
Food | Clothing | Housing | |
---|---|---|---|
Farmer | 0.40 | 0.20 | 0.20 |
Tailor | 0.10 | 0.70 | 0.20 |
Carpenter | 0.50 | 0.10 | 0.60 |
Let p1, p2
Similar equations describing the expenditures of the tailor and carpenter produce the following system of linear equations:
This system can be written as Ap=p
and A is the coefficient matrix of the system. In this context, A is called the input—output (or consumption) matrix, and Ap=p
For vectors b=(b1, b2, …, bn)
At first, it may seem reasonable to replace the equilibrium condition by the inequality Ap≤p
Hence, since the columns of A sum to 1,
which is a contradiction.
One solution to the homogeneous system (I−A)x=0
We may interpret this to mean that the society survives if the farmer, tailor, and carpenter have incomes in the proportions 25 : 35 : 40 (or 5 : 7 : 8).
Notice that we are not simply interested in any nonzero solution to the system, but in one that is nonnegative. Thus we must consider the question of whether the system (I−A)x=0
Let A be an n×n
where D is a 1×(n−1)
Observe that any input-output matrix with all positive entries satisfies the hypothesis of this theorem. The following matrix does also:
In the open model, we assume that there is an outside demand for each of the commodities produced. Returning to our simple society, let x1, x2
that is, the value of food produced minus the value of food consumed while producing the three commodities. The assumption that everything produced is consumed gives us a similar equilibrium condition for the open model, namely, that the surplus of each of the three commodities must equal the corresponding outside demands. Hence
In general, we must find a nonnegative solution to (I−A)x=d
Recall that for a real number a, the series 1+a+a2+⋯
To illustrate the open model, suppose that 30 cents worth of food, 10 cents worth of clothing, and 30 cents worth of housing are required for the production of $1 worth of food. Similarly, suppose that 20 cents worth of food, 40 cents worth of clothing, and 20 cents worth of housing are required for the production of $1 of clothing. Finally, suppose that 30 cents worth of food, 10 cents worth of clothing, and 30 cents worth of housing are required for the production of $1 worth of housing. Then the input-output matrix is
so
Since (I−A)−1
then
So a gross production of $90 billion of food, $60 billion of clothing, and $70 billion of housing is necessary to meet the required demands.
Label the following statements as true or false.
(a) Any system of linear equations has at least one solution.
(b) Any system of linear equations has at most one solution.
(c) Any homogeneous system of linear equations has at least one solution.
(d) Any system of n linear equations in n unknowns has at most one solution.
(e) Any system of n linear equations in n unknowns has at least one solution.
(f) If the homogeneous system corresponding to a given system of linear equations has a solution, then the given system has a solution.
(g) If the coefficient matrix of a homogeneous system of n linear equations in n unknowns is invertible, then the system has no nonzero solutions.
(h) The solution set of any system of m linear equations in n unknowns is a subspace of Fn
For each of the following homogeneous systems of linear equations, find the dimension of and a basis for the solution space.
(a) x1+3x2=02x1+6x2=0
(b) x1+x2−x3=04x1+x2−2x3=0
(c) x1+2x2−x3=02x1+x2+x3=0
(d) 2x1+x2−x3=0x1−x2+x3=0x1+2x2−2x3=0
(e) x1+2x2−3x3+x4=0
(f) x1+2x2=0x1−x2=0
(g) x1+2x2+x3+x4=0x2−x3+x4=0
Using the results of Exercise 2, find all solutions to the following systems.
(a) x1+3x2=52x1+6x2=10
(b) x1+x2−x3=14x1+x2−2x3=3
(c) x1+2x2−x3=32x1+x2+x3=6
(d) 2x1+x2−x3=5x1−x2+x3=1x1+2x2−2x3=4
(e) x1+2x2−3x3+x4=1
(f) x1+2x2=5x1−x2=−1
(g) x1+2x2+x3+x4=1x2−x3+x4=1
For each system of linear equations with the invertible coefficient matrix A,
(1) Compute A−1
(2) Use A−1
(a) x1+3x2=42x1+5x2=3
(b) x1+2x2−x3=5x1+x2+x3=12x1−2x2+x3=4
Give an example of a system of n linear equations in n unknowns with infinitely many solutions.
Let T: R3→R2
Determine which of the following systems of linear equations has a solution.
(a) x1+x2−x3+2x4=2x1+x2+2x3=12x1+2x2+x3+2x4=4
(b) x1+x2−x3=12x1+x2+3x3=2
(c) x1+2x2+3x3=1x1+x2−x3=0x1+2x2+x3=3
(d) x1+x2+3x3−x4=0x1+x2+x3+x4=1x1−2x2+x3−x4=14x1+x2+8x3−x4=0
(e) x1+2x2−x3=12x1+x2+2x3=3x1−4x2−7x3=4
Let T: R3→R3
(a) v=(1, 3, −2)
(b) v=(2, 1, 1)
Prove that the system of linear equations Ax=b
Prove or give a counterexample to the following statement: If the coefficient matrix of a system of m linear equations in n unknowns has rank m, then the system has a solution.
In the closed model of Leontief with food, clothing, and housing as the basic industries, suppose that the input-output matrix is
At what ratio must the farmer, tailor, and carpenter produce in order for equilibrium to be attained?
A certain economy consists of two sectors: goods and services. Suppose that 60% of all goods and 30% of all services are used in the production of goods. What proportion of the total economic output is used in the production of goods?
In the notation of the open model of Leontief, suppose that
are the input-output matrix and the demand vector, respectively. How much of each commodity must be produced to satisfy this demand?
A certain economy consisting of the two sectors of goods and services supports a defense system that consumes $90 billion worth of goods and $20 billion worth of services from the economy but does not contribute to economic production. Suppose that 50 cents worth of goods and 20 cents worth of services are required to produce $1 worth of goods and that 30 cents worth of goods and 60 cents worth of services are required to produce $1 worth of services. What must the total output of the economic system be to support this defense system?