Subspaces, span, and basis
Subspaces, span, and basis [ edit ] Main articles: Linear subspace , Linear span , and Basis (linear algebra) 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 −1 ( 0 ) of 0 (called kernel or null space), are linear subspaces of...