## Vector Space

### January 4, 2008

**What is Vector Space?**

**What is Vector Space?**

A vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms.

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**Formal Definition**

**Formal Definition**

Let *F* be a field (such as the real numbers or complex numbers), whose elements will be called scalars. A vector space over the field *F* is a set *V* together with two binary operations,

*Vector addtion:**V*×*V*→*V*denoted**v**+**w**, where**v**,**w**∈*V*, and*Scalar multiplication:**F*×*V*→*V*denoted*a***v**, where*a*∈*F*and**v**∈*V*,

satisfying the *axioms* below:

1. Vector addition is associative:

For all **u**, **v**, **w** ∈ *V*, we have **u** + (**v** + **w**) = (**u** + **v**) + **w**.

2. Vector addition is commutative:

For all **v**, **w** ∈ *V*, we have **v** + **w** = **w** + **v**.

3. Vector addition has an identity element:

There exists an element **0** ∈ *V*, called the *zero vector*, such that **v** + **0** = **v** for all **v** ∈ *V*.

4. Vector addition has inverse elements:

For all **v** ∈ V, there exists an element **w** ∈ *V*, called the *additive inverse* of **v**, such that **v** + **w** = **0**.

5. Distributivity holds for scalar multiplication over vector addition:

For all *a* ∈ *F* and **v**, **w** ∈ *V*, we have *a* (**v** + **w**) = *a* **v** + *a* **w**.

6. Distributivity holds for scalar multiplication over field addition:

For all *a*, *b* ∈ *F* and **v** ∈ *V*, we have (*a* + *b*) **v** = *a* **v** + *b* **v**.

7. Scalar multiplication is compatible with multiplication in the field of scalars:

For all *a*, *b* ∈ *F* and **v** ∈ *V*, we have *a* (*b* **v**) = (*ab*) **v**.

8. Scalar multiplication has an identity element:

For all **v** ∈ *V*, we have 1 **v** = **v**, where 1 denotes the *multiplicative identity* in *F*.

…

Some sources choose to also include two axioms of *closure* whose validity is key to determining whether a subset of a vector space is a *subspace*.

1. *V* is closed under vector addition:

If **u**, **v** ∈ *V*, then **u** + **v** ∈ *V*.

2. *V* is closed under scalar multiplication:

If *a* ∈ *F*, **v** ∈ *V*, then *a* **v** ∈ *V*.

May 12, 2008 at 3:12 am

hi there.

i just got here by chance; while updating my blog.

i’m a pure math student from argentina, struggling to get my masters degree…

i’m putting up questions and problems on my blog. you might find it interesting. check it out:

leavesandfoliations.wordpress.com

cheers

manuco.