== Examples ==
# The space [[Euclidean space|<math>\mathbb R^n</math>]] of n-tuples of real numbers is a vector space, where to add two vectors we simply add the corresponding components. The case <math>n=2</math> is exactly the case of vectors in the plane discussed above.
# The set <math>\mathbb R[x]</math> of [[polynomial|polynomials]] s with real coefficients is a vector space. If we add two polynomials together, we get another polynomial, and similarly, if we multiply a polynomial by a constant, we get another polynomial. Note that although it's also possible to multiply two polynomials and get another one, this is not part of the vector space structure: a vector space with a reasonable notion of multiplication of vectors is called an [[algebra (mathematical structure)|algebra]].
# The set of polynomials of degree less than or equal to <math>n</math> (for any <math>n \geq 0</math>) is a vector space, for the same reason.
# The set of all continuous functions on the real line is a vector space: the sum of two [[continuous|continuous functions]] is again continuous, as is the product of a continuous function with a constant.
== Properties ==
Many familiar properties of vectors in the plane carry over to the set of vector spaces. For example, just as the plane is 2-dimensional, it makes sense to talk about the dimension of any vector space (though it may be infinite, as in the case of polynomials!) Vectors in the plane can all be written in the form <math>ae_1+be_2</math>, where <math>e_1 = (1,0), e_2=(0,1)</math>, and a set elements with this same property that all vectors can be written as sums of multiples of vectors in the set is called a basis. Having a convenient basis often makes computations easier. It turns out that every finite dimensional vector space has a basis -- in basis—in fact, if we're feeling adventurous and assume the [[Axiom of Choice]], even every infinite dimensional vector space has a basis.
However, a general vector space has no notion of "distance": given a vector, there's not necessarily a way to define the length <math>|v|</math> of that vector. For example, it's not obvious how we should define the length of a polynomial or a matrix. A vector space endowed which a notion of distance is called ''normed''.
2. [[Associative property|Associativity]]: ('''u''' + '''v''') + '''w''' = '''u''' + ('''v''' + '''w''').
<br/>
3. [[Identity_ElementIdentity Element|Identity]]: There exists a '''0''' ∈ V such that '''v''' + '''0''' = '''0''' + '''v''' = '''v'''.
<br/>
4. [[Inverse]]: For all '''v''' there exists a (-'''v''') such that '''v''' + (-'''v''') =(-'''v''') + '''v''' = '''0'''.
5. [[Associative property|Associativity]]: a(b'''v''') = (ab)'''v'''.
<br/>
6. [[Identity_ElementIdentity Element|Identity]]: For 1 ∈ F (i.e, the multiplicative identity of F) it follows that 1'''v''' = '''v'''.
<br/>
==References==
Weisstein, Eric W. "Vector Space." From MathWorld--A MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/VectorSpace.html
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:vector Vector analysis]][[Category:calculusCalculus]]
[[Category:Physics]]