where is to be considered as the unknown. The solution set of a linear equation is the set of that satisfy the above constraint, or to be more precise, the pre-image . The equation is called inconsistent if no solutions exist, that is, if the pre-image is the empty set. Otherwise, the equation is called consistent.
The general solution of a linear equation has the form
is a particular solution and where
is any solution of the corresponding homogeneous problem, i.e. an element of the kernel of .
Notes. Elementary treatments of linear algebra focus almost exclusively on finite-dimensional linear problems. They neglect to mention the underlying mapping, preferring to focus instead on “variables and equations.” However, the scope of the general concept is considerably wider, e.g. linear differential equations such as
|Date of creation||2013-03-22 12:25:59|
|Last modified on||2013-03-22 12:25:59|
|Last modified by||rmilson (146)|