Real Linear Equations
Why
Linear equations are ubiquitous.
Definition
Given and , suppose
we want to find satisfying
We refer to this expression as a
real linear equation or
linear equation.
We treat each component as a
variable and we treat each component and as constants.
We call the pair the
real linear equation
constants.
The source of the terminology “linear” is by
viewing the left hand side as a function.
Define by .
We want to find to satisfy
.
Notice that is a linear real
function.
Moreover, to each linear function there exists a vector so
that .
For this reason, if we are given several
linear function , then we can
think of these as several vectors .
If we are also given for each
, then we have the vector
We can define the two-dimensional array by .
For this reason, a linear
system of equations is a pair .
A solution of a linear system of equations is
a vector satisfying the equations
Other terminology includes a
system of linear equations
or linear system or
simultaneous linear
equations