Vector Space

January 4, 2008

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

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 × VV denoted v + w, where v, wV, and
  • Scalar multiplication: F × VV denoted av, where aF and vV,

satisfying the axioms below:

1. Vector addition is associative:

For all u, v, wV, we have u + (v + w) = (u + v) + w.

2. Vector addition is commutative:

For all v, wV, we have v + w = w + v.

3. Vector addition has an identity element:

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

4. Vector addition has inverse elements:

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

5. Distributivity holds for scalar multiplication over vector addition:

For all aF and v, wV, we have a (v + w) = a v + a w.

6. Distributivity holds for scalar multiplication over field addition:

For all a, bF and vV, we have (a + b) v = a v + b v.

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

For all a, bF and vV, we have a (b v) = (ab) v.

8. Scalar multiplication has an identity element:

For all vV, 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, vV, then u + vV.

2. V is closed under scalar multiplication:

If aF, vV, then a vV.

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One Response to “Vector Space”

  1. manooko Says:

    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.


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