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[Music] Metrics is a mathematical tool that may be used to solve problems involving large data sets. One major advantage of using the metrics method is that we can express a system of equations into a single equation. And by using the matrix method, you can also save your time, money, and energy. However, the matrix method can be used to solve only linear equations. In this video, you will learn how to convert linear models into matrix form.
In matrix algebra, you will come across two concepts: scalars and vectors. A scalar is a quantity that has magnitude. It represents the size or the amount; such as, what is the length of this field? The answer will be in terms of some real number; say, for example, this field is one kilometer long. The magnitude is one kilometer, and one is a real number.
A vector, on the other hand, also has magnitude, but it gives an additional information about the direction. Vectors may be represented by using an arrow. The most widely used two-dimensional or 2D vectors may be represented in a coordinate system with the x-axis and the y-axis, like this. The arrow shows that the line moves from A to B by some magnitude, and it also has a displacement in a direction that points to the magnitude of the y-coordinate. Thus, v may be called a vector.
In economics, the law of demand says that when price decreases, quantity demanded increases, and when price increases, quantity demanded decreases, other things remaining the same. Thus, if the price of apples falls, the quantity demanded of apples will rise; and if the price of apples rises, the quantity demanded of apples will fall. Since there is an inverse relationship between quantity demanded and price, it may be said that price and quantity demanded move in opposite directions. Thus, quantity demanded may be termed as a vector.
Now, the law of demand does not give information about the magnitude of change with respect to the change in the price; that is, by how much the quantity demanded changes when the price changes. Economists use the concept of price elasticity of demand to measure the magnitude of this change. Elasticity of demand does not tell anything about the direction, but it has a magnitude; hence, it may be called a scalar.
In mathematical language, a vector is a special type of matrix that has only one row or one column. If a matrix has one row, it is referred to as a row vector. If a matrix has one column, it is referred to as a column vector. Thus, r = 2 3 4 is a row vector with a collection of numbers 2, 3, and 4 in one row. c = 2 3 4 is a column vector with a collection of numbers 2, 3, and 4. A number of such row vectors and column vectors taken together forms a matrix. Thus, a matrix may be defined as a collection of vectors.
Us now proceed to convert linear equations or linear models into the matrix form. Suppose you have a set of three equations. The observations are: there are three variables, x, y, and z. The position of the variables in all the three equations are not identical. Variable y is written first in the second equation. Variable z is missing in the second equation, and variable y is missing in the third equation.
To convert the system of three equations to matrix form, we first arrange the equations by placing the variables in the same order as follows. The missing variables are given a coefficient 0, and a variable without any coefficient is given a coefficient of 1. Second, we separate the coefficients of each equation in rows and columns and give a name to this matrix. Let us say Z is a three by three matrix of all the coefficients from the given system of three equations. The rows represent the three equations, and the columns represent the three variables placed in the order of x, y, and z. If you remember, the second equation had y written first, so care must be taken while arranging and writing the coefficient of y. The coefficient of y is 3 and is written in the second column. Since z and y are missing in the second and third equations respectively, the coefficient for z and y is written as zero.
Next, the variables and constants are written as a column vector. Let us name the vector of variables as v = x y and z. The constants on the right-hand side of the equations may also be written as a column vector. Let us call it t = 40 43 and 38. Now, the system of three linear equations would look somewhat like this in matrix form. In shorthand, this may be written as Zv = t, where Z is a matrix of order three by three, v is a matrix of order three by one, and t is a matrix of order three by one. Therefore, three equations have been compressed to one equation. The only thing to be kept in mind is that unlike single equations, v is not a single variable but a collection of three variables. [Music] You