向量空間子空間定義
A subset
of a vector space $V is called a subspace of is is itself a vector space under the addition and scalar multiplication defined on .
向量空間的子空間需要同時滿足元素來自於原向量空間且向量加法運算與純量乘法運算與原空間相同。
如何判斷向量空間的某子集合是不是其子空間
(重要)Theorem:
If
is a nonempty subset of a vector space , then is a subspace of if and only if the following conditions hold: <a>
<b>
and is a scalar, 即若向量空間的非空子集滿足向量加法和純量乘法的封閉性則其為該向量空間的子空間。
Proof:
(
)
W is itself a vector
Axiom1 and Axiom6 hold. i.e. condition<a> and <b> hold.
(
)
<a> and <b> are Axiom1 and Axiom6, respectively.
Axiom1 and Axiom6 hold. Also, Axiom2,3,7,8,9 and 10 hold for inherited property.
For condition<b>,
is scalar, By taking k = 0 and substituting into (1), we have:
(by theorem) By taking k = -1 and substituting into (1), we have:
(by theorem)
Axiom4 and Axiom5 are hold.
一般判斷是否為子空間步驟:
先判斷該子集合内有無零向量，再判斷兩個封閉性。
or not or not or not
某向量空間的子空間的交集仍為其子空間
Any intersection of subspace of a vector space
is a subspace of .
線性組合
Def:
If
is a vector in a vector space V, then u is said to be a linear combination of the vector in , if can be expressed in the form where are scalars. These scalars are called the cofficient of the linear combination.
Theorem: (包含向量空間子集的最小子空間是該子集內向量的線性組合形成之子空間) >if
Def of Span:
The subspace
of a vector space that is formed from all possible linear combinations of the vectors in a nonempty set is called the span of and defined by . i.e. Span(
) =
Theorem:
The solution set of a homogeneous linear system
in unknowns is a subspace of
Proof:
let
be the solution set of , then is a scalar
by theorem,