section_path: Vector spaces/Subspaces, span, and basis
section_name: Subspaces, span, and basis
paragraph_text: Subspaces, span, and basis
  
  The study of those subsets of vector spaces that are in themselves vector spaces under the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces. More precisely, a linear subspace of a vector space V over a field F is a subset W of V such that u + v and a u are in W, for every u, v in W, and every a in F. (These conditions suffice for implying that W is a vector space.)
  
  For example, given a linear map T : V → W, the image T (V) of V, and the inverse image T (0) of 0 (called kernel or null space), are linear subspaces of W and V, respectively.
  
  Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums
  
  where v, v,..., v are in S, and a, a,..., a are in F form a linear subspace called the span of S. The span of S is also the intersection of all linear subspaces containing S. In other words, it is the smallest (for the inclusion relation) linear subspace containing S.
  
  A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vectors is linearly independent if the only way to express the zero vector as a linear combination of elements of S is to take zero for every coefficient a.
  
  A set of vectors that spans a vector space is called a spanning set or generating set. If a spanning set S is linearly dependent (that is not linearly independent), then some element w of S is in the span of the other elements of S, and the span would remain the same if one were to remove w from S. One may continue to remove elements of S until getting a linearly independent spanning set. Such a linearly independent set that spans a vector space V is called a basis of V. The importance of bases lies in the fact that they are simultaneously minimal generating sets and maximal independent sets. More precisely, if S is a linearly independent set, and T is a spanning set such that S ⊆ T, then there is a basis B such that S ⊆ B ⊆ T.
  
  Any two bases of a vector space V have the same cardinality, which is called the dimension of V; this is the dimension theorem for vector spaces. Moreover, two vector spaces over the same field F are isomorphic if and only if they have the same dimension.
  
  If any basis of V (and therefore every basis) has a finite number of elements, V is a finite-dimensional vector space. If U is a subspace of V, then dim U ≤ dim V. In the case where V is finite-dimensional, the equality of the dimensions implies U = V.
  
  If U and U are subspaces of V, then
  
  where U + U denotes the span of U ∪ U.