Conditions Giving Rise To Indeterminacy
In linear systems, indeterminacy occurs if and only if the number of independent equations (the rank of the augmented matrix of the system) is less than the number of unknowns and is the same as the rank of the coefficient matrix. For if there are at least as many independent equations as unknowns that will eliminate any stretches of overlap of the equations' surfaces in the geometric space of the unknowns (aside from possibly a single point); and if the rank of the augmented matrix exceeds (necessarily by one if at all) the rank of the coefficient matrix then the equations jointly contradict each other.
Read more about this topic: Indeterminate System
Famous quotes containing the words conditions, giving and/or rise:
“Mankind always sets itself only such tasks as it can solve; since, looking at the matter more closely, we will always find that the task itself arises only when the material conditions necessary for its solution already exist or are at least in the process of formation.”
—Karl Marx (1818–1883)
“One promises much, to avoid giving little.”
—Luc, Marquis de Vauvenargues (1715–1747)
“And now you try
Your handful of notes;
The clear vowels rise like balloons.”
—Sylvia Plath (1932–1963)