basis(基底)
Def of basis:
If
is any vector space and is a subset of , then is called a basis for if the following two conditions hold: <a>
is linearly independent <b>
spans 形成basis的兩個條件，一個是基底集合內向量互相獨立，另一個是該基底集合可以編織出整個向量空間。
Theorem(基底對於向量空間的意義,與上邊def相同意思):
If
is a nonempty subset of a vector space , then is a basis for if and only if every vector in can be expressed in the form in exactly one way, where are scalars.
proof:
(正向)
is a basis for
such that
Suppose that the expression of
in terms of is not unique. i.e. such that
By
i.e. the expression of
in terms of is unique.
(反向)
such that
Suppose that
is L.D., then not all zeros, such that By
By Equ(3) and Equ(5), we have the expression of
in terms of is not unique(contradictory, the expression is only in one way)
is L.I. i.e.
is a basis for
Coordinate(坐標)
If
is a basis for and in exactly one way, , then these scalars are called the coordinates of relative to the basis . The vectors in constructed from these coordinates is called the coordinate vector of relative to , denoted by or