Meaning, geometric representation, and types of vectors

Vector operations

 

 

Lecture

Matrices and operations on matrices. Determinants. Basic properties. Rank of the matrix.

 

 

 

Two of the importantproperties of scalar multiplicationof matrices are the following.

 

 

 

Lecture

System of linear algebraic equations and methods for their solutions (Formula Cramer, Gauss methods).

 

 

Lecture

The use of elements of linear algebra in the economy. Leontief model of a diversified economy.

The use of elements of linear algebra in the economy. Leontief model of a diversified economy.

Macroeconomics operation of a diversified economy requires a balance between the different branches. Each sector on the one hand, is prizvoditelem, and on the other - the consumer products produced by other industries. There is quite a difficult task of calculating the relationship between the branches through the production and consumption of products of different types. This problem was first formulated in the form of a mathematical model in 1936 in the writings of well-known American economist Leontief, who tried to analyze the causes of the economic depression of the United States 1929-1932. This model is based on matrix algebra and matrix analysis using the device.

Balance sheet ratio

For simplicity, we assume that the manufacturing economy is a sphere n branches, each of which produces a homogeneous product. To secure its production every industry needs from other sectors (industrial consumption). Typically the process of production is considered for a certain period of time; in some cases, such a unit is the year.

We introduce the following notation:

x_i - total output of i-th branch (its gross output);

the volume of product i-th industry consumed j-th industry in the production volume

of products xj;

the volume of product i-th branch, intended for sale (consumption) in the non-manufacturing

sector, or the so-called end-use product. This includes personal consumption of the citizens,

satisfaction of social needs, the content of the state institutions, etc.

 

The balance principle of communication in various industries is that the gross output of the i-th sector should be equal to the sum of consumption volumes in the industrial and non-industrial sectors. In its simplest form (the hypothesis of linearity, or simple addition) balance relations are of the form

 

This equation (16.2) is called the balance of relations.

Since the products of different industries have different dimensions, we will in the future to keep in mind the value balance.