Argument about defining basic variable in matrix being
trivial
Definition of basic variable and the properties of it derived
from its definition
In a matrix that represents a set of equations about some relationships
between some variables x1, x2, … xn, basic variables, among all variables
between which relationships are represented in the matrix, is defined as the
variable of the leading entry, where the leading entry of a row is defined as
the left most non-zero element in a row.
From such definition of basic variables in matrixes, we shall see some
specific limits of defining basic variables:
1. Basic variables are defined upon the definition of leading entry, who’s
defined as the left most non-zero element in a row. Since a matrix is an
alternative and one can surely argue, better, representation of a set of
equations of some variables which represents numerically the
relationships between the variables, then according to the commutative
law of additions, one can re-write the equations in orders of the
variables arbitrary, since it doesn’t change the relationships which it’s
representing. For example, the relationship 2x+y+3z=4 can be rewritten arbitrarily for one’s own purpose in other forms like
y+2x+3z=4, 3z+y+2x=4, etc.
2. The definition of basic variables is followed by the definition of free
variables as a consequence of defining some variables in the
relationships as basic variables. The free variables are defined as
variables contained in the relationships that are not basic variables.
Therefore, by defining basic variables in some relationships, we would
also corresponding defining other variables left to be the free variables
at the same time. Alike for any other two or more variables, basic
variables shall be expressed in terms of the free variables. (Alike any
variables can be expressed in terms of some other variables.)
Interpretations of such properties of basic variables
As discussed, since a matrix is an alternative representation of a set of
equations representing some relationships between some variables, for such
equations, one can arbitrarily change the order of variables in which the
equation is written without changing the relationship it was representing.
Correspondingly, when the set of equation is expressed in from of a matrix,
one shall then change the order of the coefficients of the variables as long as
such change is done for each row in the matrix, since in a matrix, for better
and easier representation’s sake, the order of the coefficients of variable
written is the same for every row, that is, for each column of a matrix, the
numbers are the coefficients of the same variable.
By doing so as said above, one shall then arbitrarily change the position of
variables, thus one can arbitrarily change the “left most non-zero element in a
row” without changing the relationships, as long as the exact same change of
the order of variables is done for every row in the matrix. Thus, according to
the definition of basic variables, which defines basic variables as the variable
of the left most non-zero element in a row, one can then now arbitrarily
change which variable among all shall be defined as the basic variables and
which shall be defined as the free variables, correspondingly.
As discussed in the previous session, since all variables within a matrix is
defined as either basic variables or free variables, thus for each relationship
between all variables in the matrix, there is a corresponding relationship
between basic variables and free variables, and such relationship remains to
exist or being none regardless of the choice of which variables among all are
defined as the basic or free variables. That is, one shall arbitrarily change
which variables are defined as basic or free variables, but such relationships
between them, although may change accordingly, remains to exist if such
relationships exist before the change of the choice of basic and free variables.
The ideas & facts represented by defining basic & free
variables
The definitions of basic and free variables, as represented and interpreted
above, don’t separate all variables in a matrix into certain categories since
one can arbitrarily change which variables among all shall be defined as
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which kinds of variable, thus each variable in a matrix shall then be defined
as either kind of variable: basic or free variable, for one’s purpose, at one’s
will. Meanwhile, since the basic and free variables are defining the
previously existed all variables in the matrix to be one of them, thus the
relationships represented in the matrix remain the same, since basic and free
variables are only giving already-exist variables new name for putting them
into categories, after defining the variables to be basic or free variables,
there’re always corresponding relationships between them, if there exists any
relationships between the already-exist variables in the matrix when the
matrix was defined and those relationships were maintained within.
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