Vector space ↩

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created: 2021-11-06 00:10:52
modified: 2022-01-10 04:13:04

The operations must satisfy the following properties and :

1. is an abelian group under (identity is denoted as )
3. The two operations are related by:

Motivation with

Let's take as a concept, the plane of real numbers: . It is the set of ordered pairs .

We can define the addition then like this:

This is associative, acts as an identity and is the inverse becomes a group, an abelian group to be specific since the addition is clearly commutative.

Next, we define the next operation where we multiply an element of with a real number to get another element in . For and :

Examples

• Of course , but it is true for arbitrary :
• The field is a vector space over itself. The vector addition is just addition in and scalar multiplication is just multiplication in .
• : -tuples of elements of . This is a vector space over just as is a real vector space
• Let be the set of all continuous functions . We define
  Where and Since both functions are continuous, will be continuous as well and is continuous, so becomes a real vector field.