Positive Roots and Simple Roots
Given a root system Φ we can always choose (in many ways) a set of positive roots. This is a subset of Φ such that
- For each root exactly one of the roots, – is contained in .
- For any two distinct such that is a root, .
If a set of positive roots is chosen, elements of are called negative roots.
An element of is called a simple root if it cannot be written as the sum of two elements of . The set of simple roots is a basis of with the property that every vector in is a linear combination of elements of with all coefficients non-negative, or all coefficients non-positive. For each choice of positive roots, the corresponding set of simple roots is the unique set of roots such that the positive roots are exactly those that can be expressed as a combination of them with non-negative coefficients, and such that these combinations are unique.
Read more about this topic: Root System
Famous quotes containing the words positive, roots and/or simple:
“I believe, as Maori people do, that children should have more adults in their lives than just their mothers and fathers. Children need more than one or two positive role models. It is in your children’s best interest that you help them cultivate a support system that extends beyond their immediate family.”
—Stephanie Marston (20th century)
“There is nothing but is related to us, nothing that does not interest us,—kingdom, college, tree, horse, or iron show,—the roots of all things are in man.”
—Ralph Waldo Emerson (1803–1882)
“Armies, for the most part, are made up of men drawn from simple and peaceful lives. In time of war they suddenly find themselves living under conditions of violence, requiring new rules of conduct that are in direct contrast to the conditions they lived under as civilians. They learn to accept this to perform their duties as fighting men.”
—Gil Doud, U.S. screenwriter, and Jesse Hibbs. Walter Bedell Smith (Himself)