Dimension(向量空間維度)
Theorem(基底向量個數恆定):
Let
be a finite-dimensional vector space and let be a basis for , then <a>If a set in
that has more than vectors, then it is linearly dependent. <b>If a set in
that has fewer than vectors, then it does not span . 基底向量的個數代表了維度的個數，亦表示向量空間在該維度上的特性，所以有以上結論。
Theorem(向量空間的所有基底內向量數相同):
All bases for a finite-dimensional vector space have the same number of vectors.
Def of dimension:
The dimension of a finite-dimensional vector space
, denoted is defined to be the number of vectors in a basis for . In addition, the zero vector space is defined to have dimension zero.
Ex1:
Let
be the all n-by-n symmetric matrix, then find: <a> dim(V)
dim(v) =
對稱矩陣的性質只體現在主軸及一半區域
<b> a basis for V
由一般矩陣基底推得
EX2:
Theorem(Plus minus theorem 用於建構基底的定理):
Let
be a nonempty set of vectors in a vector space <a>If
is L.I. and if ,v , then is still L.I. 當向量空間內某些向量無法被某集合編織出時，可以通過聯集的形式加入該向量，不影響線性獨立的特性且愈發接近基底。
<b> If
and is expressible as a linear combination of other vectors in , then
Theorem(對於向量數等於維度的集合，判斷是否為基底的方法):
let
be a dimensional vector space, let be a set in with exactly vectors. Then is a basis for if spans or is linearly independent.
Proof:
If S is L.I. then suppose that
By plus theorem,
such that is L.I. But
(Contradictory, By theorem, is L.D.)
If
then suppose that is L.D.
and By minus theorem,
(Contradictory, )
is L.I.
Theorem:
If
is a subspace of a finite dimensional vector space <a>
is finite-dimensional <b>
<c>
if and only if