Vector space: Difference between revisions

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'''Vector spaces''' are an abstract [[mathematics|mathematical]] construct with many important applications in the  
'''Vector spaces''' are an abstract [[mathematics|mathematical]] construct with many important applications in the  
[[CZ:Workgroups#Natural_Sciences|Natural Sciences]], in particular in [[physics]] and other areas of mathematics.
[[CZ:Workgroups#Natural_Sciences|Natural Sciences]], in particular in [[physics]] and other areas of mathematics.
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===Differential equations===
===Differential equations===
[[Category:CZ Live]]
[[Category:Mathematics Workgroup]]

Revision as of 17:44, 15 November 2007

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Vector spaces are an abstract mathematical construct with many important applications in the Natural Sciences, in particular in physics and other areas of mathematics. Some vector spaces make sense somewhat intuitively, such as the space of 3D vectors in standard Euclidean space, and the language that we use when talking about these intuitive spaces has been taken to describe the more abstract notion as well. For example, we know how to add vectors and multiply them by real numbers (scalars) in , and these notions of vector addition and scalar multiplication are defined in a more general sense (as we will see below).

Vector spaces are important because many different mathematical objects that at first glance seem unrelated in fact share a common structure. By defining this structure and proving things about it in general, we are then able to apply these results to each specific case without having to re-prove them each time. Besides vectors in that are relatively easy to visualize, we can make a vector space out of for any natural number ; or the complex plane or powers of it, ; or polynomials of degree .

No matter what vector space you have to work with though, it is often useful to keep a picture of either 2D or 3D space in mind. This helps when thinking of things such as orthogonal polynomials or matrices.


Definition

A vector space over a field is a set that satisfies certain axioms (see below) and which is equipped with two operations, vector addition and scalar multiplication. Vector addition is defined as a map

that takes the ordered pair to the vector . Here represents the Cartesian product between sets. Scalar multiplication is defined in a similar way, as a map

that takes the ordered pair to the vector . Note that frequently the dot representing scalar multiplication is omitted, the result being written simply as instead. This is especially common when an inner product will also be defined on the vector space, with the dot then representing the inner product between two vectors. It is important to keep in mind the distinction between scalar multiplication, which multiplies one vector by a scalar, and an inner or scalar product, that combined two vectors to yield a scalar.

Axioms of a vector space

Let be a set, , , and elements of that set, and and scalar elements of a field, . Then is a vector space if the following axioms hold true for all choices of

1. is closed under addition
The vector is also an element of . This is automatically satisfied when the addition operation is defined as being injective as it was above. Care must be taken however if is a subset of some larger set and , as is often the case when looking at subspaces.
2. Addition is commutative
The order in which two vectors are added does not affect the result, .
3. Addition is associative
. This means that even though addition is strictly defined as a binary operation, the object is well defined.
4. An additive identity exists in
Labeled , the additive identity or zero vector satisfies .
5. The additive inverse exists in
A vector can be found such that .
6. is closed under scalar multiplication
The vector is itself an element of .
7. Scalar multiplication is distributive over addition in
. It is important to note that the addition occurring on the left-hand side of this equality is a 'different operation' from the addition on the right-hand side. While the latter is vector addition as defined above, the former is the addition operation defined on the field .
8. Vector addition is distributive over scalar multiplication
. In this case vector addition takes place on both sides of the equality.
9. Scalar multiplication is associative
. This means that the algebraic structure of the underlying field is preserved. Note that the left-hand side of this equality contains two subsequent applications of the scalar multiplication defined above, while the right-hand side contains one scalar multiplication as defined in (that of ), followed by scalar multiplication with the vector .
10. The multiplicative identity of provides a scalar multiplicative identity
, where is the multiplicative identity of the field .

These axioms can be expressed concisely in mathematical notation as follows:

Some important theorems

Examples of vector spaces

Applications of vector spaces

Classical mechanics

Quantum mechanics

Differential equations