Home/Algebra/Vector spaces/Basis and dimension18. Vector spacesBasis and dimensionLevel: intermedioIn 18. Vector spacesFormulaCopy LaTeXdim(V)=#{vectores on una base}\dim(V)=\#\{\text{vectores on una base}\}dim(V)=#{vectores on una base}Tagsalgebraspaces vectoralgebra formulalinearDetailA basis spans the space and is linearly independent.Explore the ideaFound an error? Contact us.Related formulasRelatedSpanspan(v1,…,vk)={∑icivi}\operatorname{span}(v_1,\ldots,v_k)=\{\sum_i c_iv_i\}span(v1,…,vk)={i∑civi}Open formulaOpen in sectionRelatedLinear independence∑icivi=0⇒c1=⋯=ck=0\sum_i c_iv_i=0\Rightarrow c_1=\cdots=c_k=0i∑civi=0⇒c1=⋯=ck=0Open formulaOpen in sectionRelatedCoordinates in a basisv=∑icibi⇒[v]B=(c1,…,cn)Tv=\sum_i c_ib_i\Rightarrow[v]_\mathcal B=(c_1,\ldots,c_n)^Tv=i∑cibi⇒[v]B=(c1,…,cn)TOpen formulaOpen in section