Matrix algebra can enable us to do many things.
1) It provides a compact way of writing an equation system, even an extremely large one.
2) It leads to a way of testing the existence of solution by evaluation of a determinant
3) It gives a method of finding that solution (if it exists)
However, one slight catch is that matrix algebra is applicable only to linear-equation system. In many cases, even if some sacrifice of realism is entailed by the assumption of linearity, an assumed linear relationship can produce sufficiently close approximation to an actual nonlinear relationship to warrant its use.
In other cases, while preserving the non-linearity in the model, we can effect a transformation of variables so as to obtain a linear relation to work with. For example, the nonlinear function,
can be readily transformed, by taking the logarithm on both sides as:
log y = log a + b logx
Matrices and Vectors
In general, a system of m linear equations in n variables
can be arranged into such a format:
(11.1)
...............................................................
There are essentially three types of ingredients in the equation system 11.1. The first is set of coefficients
; the second is the set of variables ; and the last is the set of constant terms 
. If we label them, respectively, as A, x, and d, then we have:
As a shorthand device, the array in matrix A can be written more simply as:
Some matrices may contain only one column, such as x and d above. Such matrices are given the special name column vectors. If we arrange the vector x in horizontal way, though, there would result a 1 x n matrix, which is called row vector. A row vector is often distinguished from a column vector by the use of a prime symbol, X'.
A vector, whether row or column, is merely an ordered n-tuple, and as such it may sometimes be interpreted as a point in an n-dimensional space. With the matrices defined above, we can express the equation system in 11.1 as,
Ax = d
However, the equation Ax = d prompts at least two questions. How do we multiply two matrices A and x? What is meant by the equality of Ax and d? Since matrices involve whole block of numbers, the familiar algebraic operations defined for single numbers are not directly applicable, and there is a need for a new set of operational rules, which would be discussed in Post 12 onwards.
1) It provides a compact way of writing an equation system, even an extremely large one.
2) It leads to a way of testing the existence of solution by evaluation of a determinant
3) It gives a method of finding that solution (if it exists)
However, one slight catch is that matrix algebra is applicable only to linear-equation system. In many cases, even if some sacrifice of realism is entailed by the assumption of linearity, an assumed linear relationship can produce sufficiently close approximation to an actual nonlinear relationship to warrant its use.
In other cases, while preserving the non-linearity in the model, we can effect a transformation of variables so as to obtain a linear relation to work with. For example, the nonlinear function,
can be readily transformed, by taking the logarithm on both sides as:
log y = log a + b logx
Matrices and Vectors
In general, a system of m linear equations in n variables
...............................................................
There are essentially three types of ingredients in the equation system 11.1. The first is set of coefficients
As a shorthand device, the array in matrix A can be written more simply as:
Some matrices may contain only one column, such as x and d above. Such matrices are given the special name column vectors. If we arrange the vector x in horizontal way, though, there would result a 1 x n matrix, which is called row vector. A row vector is often distinguished from a column vector by the use of a prime symbol, X'.
A vector, whether row or column, is merely an ordered n-tuple, and as such it may sometimes be interpreted as a point in an n-dimensional space. With the matrices defined above, we can express the equation system in 11.1 as,
Ax = d
However, the equation Ax = d prompts at least two questions. How do we multiply two matrices A and x? What is meant by the equality of Ax and d? Since matrices involve whole block of numbers, the familiar algebraic operations defined for single numbers are not directly applicable, and there is a need for a new set of operational rules, which would be discussed in Post 12 onwards.
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