# Vector Space

A vector space is a set that is closed under finite vector addition and scalar multiplication. The basic example is *n*-dimensional Euclidean space.

Vector space is a college-level concept that would be first encountered in a linear algebra course.

### Examples

Euclidean Space: | Euclidean space of dimension n is the space of all n-tuples of real numbers which generalizes the two-dimensional plane and three-dimensional space. |

### Prerequisites

Banach Space: | A Banach space is a vector space that has a complete norm. Banach spaces are important in the study of infinite-dimensional vector spaces. |

Hilbert Space: | A Hilbert space is a vector space that has a complete inner product. Hilbert spaces are important in the study of infinite-dimensional vector spaces. |

Matrix: | A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, for every linear transformation, there exists exactly one corresponding matrix, and every matrix corresponds to a unique linear transformation. The matrix is an extremely important concept in linear algebra. |

Scalar: | A scalar is value (such as a measurement) that has only magnitude but not direction. This contrasts with a vector, which has direction as well as magnitude. |

Tangent Space: | A tangent space is a vector space of all possible tangent vectors to a point on a manifold. |

Vector: | (1) In vector algebra, a vector mathematical entity that has both magnitude (which can be zero) and direction. (2) In topology, a vector is an element of a vector space. |