1.6 Applications of linear systems (线性方程组的应用)

本文为《Linear algebra and its applications》的读书笔记

目录

• A Homogeneous System in Economics
• Balancing Chemical Equations
• Network Flow 网络流

A Homogeneous System in Economics

The system of 500 equations in 500 variables, mentioned in this chapter’s introduction, is now known as a Leontief “input–output” (or “production”) model. (列昂惕夫“投入-产出” / “生产”模型). Section 2.6 will examine this model in more detail, when more theory and better notation are available. For now, we look at a simpler “exchange model (交易模型),” also due to Leontief.

Suppose a nation’s economy is divided into many sectors, such as various manufacturing, communication, entertainment, and service industries. Suppose that for each sector we know its total output for one year(年度总产出) and we know exactly how this output is divided or “exchanged” among the other sectors of the economy. Let the total dollar value(总货币价值) of a sector’s output be called the price(价格) of that output. Leontief proved the following result.

• There exist equilibrium prices (平衡价格) that can be assigned to the total outputs of the various sectors in such a way that the income of each sector exactly balances its expenses.

The following example shows how to find the equilibrium prices.

EXAMPLE 1
Suppose an economy consists of the Coal(煤炭), Electric (power), and Steel sectors, and the output of each sector is distributed among the various sectors as shown in Table 1, where the entries in a column represent the fractional parts of a sector’s total output.

Denote the prices (i.e., dollar values) of the total annual outputs of the Coal, Electric, and Steel sectors by $p_C$pC​, $p_E$pE​, and $p_S$pS​, respectively. If possible, find equilibrium prices that make each sector’s income match its expenditures.

SOLUTION



Any (nonnegative) choice for $p_S$pS​ results in a choice of equilibrium prices.

Balancing Chemical Equations

Network Flow 网络流

A $network$network consists of a set of points called $junctions$junctions, or $nodes$nodes, with lines or arcs called $branches$branches connecting some or all of the junctions. The direction of flow in each branch is indicated, and the flow amount (or rate) is either shown or is denoted by a variable.

The basic assumption of network flow is that the total flow into the network equals the total flow out of the network and that the total flow into a junction equals the total flow out of the junction. (the flow is “conserved”(守恒) at each junction)

EXAMPLE 2
The network in Figure 2 shows the traffic flow (in vehicles per hour) over several one-way streets in downtown Baltimore during a typical early afternoon. Determine the general flow pattern for the network.

SOLUTION

Also, the total flow into the network equals the total flow out of the network, which simplifies to $x_3 = 400$x3​=400.