向量空間定義

Def:

Let be an arbitary nonempty set of object on which two operations called addition and scalar multiplication, are defined so that for each pari of object in , there is a unique object in , and a unique object in , is a scalar.

If the following axioms are satisfied by all object in and all scalar , then we called a vector space and the object in vectors:

  1. If in then in (closure property)

  2. (加法交換律)

  3. (加法結合律)

  4. called a zero vector for such that (這邊的零向量不一定是數值意義上的零，代表零向量之意，如後邊EX8)

  5. called a negative of u such that (向量空間内所有元素都存在加法運算反元素，這邊的零同樣代表zero vector的意思，如後EX8)

  6. if in , is scalar, then in (closure property)

  7. (純量乘法對向量加法分配率)

  8. (純量乘法對純量加法分配率)

  9. (純量乘法結合律)

  10. (純量1乘法不改變向量特徵)

Rewrite Axiom 4:

There exist an element in a vector space , denoted by , called the zero vector, such that

Rewrite Axiom 5:

For each element in a vector space , there exist an element in such that , where y is called the (addition) inverse of .

Tip: For the uniqueness theorem, we denote , called the negative of , to be the (addition) inverse of .

Lemma:

<a> The zero vector in a vector space is unique. 向量空間内零向量唯一

<b> The (addition) inverse of in a vector space is unique. 向量空間内某元素的加法反元素唯一

proof of <a>:

Suppose that and are zero vector of , then

---by def of zero vector and is a zero vector

--- addition is commutative

---def of zero vector and is a zero vector

i.e.

The zero vector in a vector space is unique.

proof of <b>:

Suppose that and are inverse of , then

--- def of addition inverse

i.e.

The (addition) inverse of in a vector space is unique.

常見向量空間

實數向量空間(Euclidean n-space,歐幾里得空間)

Define"

Then, , defined above,is a vector space with:

Zero vector:

Inverse of u:

零向量空間

Define: , k is a scalar,then V, defined above, is a vector space, called a zero vector space.

無窮維向量空間

Define:

Then , defined above, is a infinitely dimension vector space.

多項式向量空間

Let

Define:

Then defined above, is a vector space with

Zero Vector:

Inverse of

m-by-n矩陣向量空間

Let

is a scalar

Define:

Then is a vector space with

Zero Vector:

Inverse of

實值函數向量空間

Let be the set of real-valued functions that are defined at each in the interval

i.e.

is a scalar, define:

Then , defined above, is a vector space with

Zero Vector:

Inverse of

一個特殊的向量空間(EX8)

Let

is a scalar, we define:

Then , defined above is a vector space with

Zero Vector:

Inverse of

不是向量空間的例子

向量空間内向量與純量1的乘積仍需為本身

Let

Define:

It's not a vector space, cuz:

這種重新定義的純量乘法運算會使得向量失去一個維度的分量，當其與純量1相乘時不能得到其本身。

向量空間内向量與零向量的和應為本身

Let and is a scalar

Define:

)

It's not a vector space.我們可以看出這邊定義的向量加法會導致二維的向量變爲一維的向量，顯然對於：不滿足，以下證明：

suppose that it's a vector space with is the zero vector of it.

, we have:

i.e. we have:

矛盾，故這不是一個向量空間。

向量空間内元素的幾個性質

Theorem:

Let be a vector space, and is a scalar, then

<a> 純量0和向量的乘積為零向量

<b> 純量(-1)和向量的乘積為其inverse

<c> 零向量與純量的乘積仍為其本身

<d> if then or

proof of <a>:

---by 分配律

---by def of zero vector

---有加法運算反元素，可消去

proof of <b>:

---by 分配律

---by <a>

---有加法運算反元素，可消去