Home/Algebra/Vector spaces/Rank18. Vector spacesRankLevel: intermedioIn 18. Vector spacesFormulaCopy LaTeXrank(A)=dim(Col(A))\operatorname{rank}(A)=\dim(\operatorname{Col}(A))rank(A)=dim(Col(A))Tagsalgebraspaces vectoralgebra formulalinearDetailNumber of independent directions produced by A.Explore the ideaFound an error? Contact us.Related formulasRelatedRouché–Capelli theoremrank(A)=rank([A∣b])=norrank(A)=rank([A∣b])<norrank(A)≠rank([A∣b])\operatorname{rank}(A)=\operatorname{rank}([A|b])=n \quad\text{or}\quad \operatorname{rank}(A)=\operatorname{rank}([A|b])<n \quad\text{or}\quad \operatorname{rank}(A)\neq\operatorname{rank}([A|b])rank(A)=rank([A∣b])=norrank(A)=rank([A∣b])<norrank(A)=rank([A∣b])Open formulaOpen in sectionRelatedRank–nullity theoremn=rank(A)+nullity(A)n=\operatorname{rank}(A)+\operatorname{nullity}(A)n=rank(A)+nullity(A)Open formulaOpen in sectionRelatedSVDA=UΣVTA=U\Sigma V^TA=UΣVTOpen formulaOpen in section