Extending To A Basis
Let S be a subset of a vector space V. To extend S to a basis means to find a basis B that contains S as a subset. This can be done if and only if S is linearly independent. Almost always, there is more than one such B, except in rather special circumstances (i.e. S is already a basis, or S is empty and V has two elements).
A similar question is when does a subset S contain a basis. This occurs if and only if S spans V. In this case, S will usually contain several different bases.
Read more about this topic: Basis (linear Algebra)
Famous quotes containing the words extending and/or basis:
“The radiance was that of the full, setting, and blood-red moon, which now shone vividly through that once barely- discernible fissure,... extending from the roof of the building, in a zigzag direction, to the base. While I gazed, this fissure rapidly widened.”
—Edgar Allan Poe (1809–1849)
“The cultivation of literary pursuits forms the basis of all sciences, and in their perfection consist the reputation and prosperity of kingdoms.”
—Marquês De Pombal (1699–1782)