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Examples of vector spaces
This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis.
Notation. Let F denote an arbitrary field such as the real numbers R or the complex numbers C.
The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see the third axiom in the Vector space article). Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one.
The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L. (Incidentally, the null space of L is a zero space if and only if L is injective.)
The next simplest example is the field F itself. Vector addition is just field addition, and scalar multiplication is just field multiplication. This property can be used to prove that a field is a vector space. Any non-zero element of F serves as a basis so F is a 1-dimensional vector space over itself.
The field is a rather special vector space; in fact it is the simplest example of a commutative algebra over F. Also, F has just two subspaces: {0} and F itself.
A basic example of a vector space is the following. For any positive integer n, the set of all n-tuples of elements of F forms an n-dimensional vector space over F sometimes called coordinate space and denoted Fn. An element of Fn is written
where each xi is an element of F. The operations on Fn are defined by
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Examples of vector spaces
This page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis.
Notation. Let F denote an arbitrary field such as the real numbers R or the complex numbers C.
The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see the third axiom in the Vector space article). Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one.
The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L. (Incidentally, the null space of L is a zero space if and only if L is injective.)
The next simplest example is the field F itself. Vector addition is just field addition, and scalar multiplication is just field multiplication. This property can be used to prove that a field is a vector space. Any non-zero element of F serves as a basis so F is a 1-dimensional vector space over itself.
The field is a rather special vector space; in fact it is the simplest example of a commutative algebra over F. Also, F has just two subspaces: {0} and F itself.
A basic example of a vector space is the following. For any positive integer n, the set of all n-tuples of elements of F forms an n-dimensional vector space over F sometimes called coordinate space and denoted Fn. An element of Fn is written
where each xi is an element of F. The operations on Fn are defined by