An　r-uniform　hypergraph　is　linear　if　every　two　edges　intersect　in　at　most　one　vertex.　Let　Kr+1　be　a　complete　graph　with　r　+　1　vertices.　The　r-uniform　hypergraph　Kr++1　is　obtained　from　Kr+1　by　enlarging　each　edge　of　Kr+1　with　r　-　2　new　vertices　disjoint　from　V　(Kr+1)　such　that　distinct　edges　of　Kr+1　are　enlarged　by　distinct　vertices.　Let　H　be　a　Kr++1-free　linear　r-uniform　hypergraph　with　n　vertices.　In　this　paper,　we　prove　that　when　n　is　sufficiently　large,　the　spectral　radius　\rho　(H)　of　the　adjacency　tensor　of　H　is　no　more　than　nr　,　i.e.,　\rho　(H)　\leq　nr　,　with　equality　if　and　only　if　r|　n　and　H　is　a　transversal　design,　where　the　transversal　design　is　the　balanced　r-partite　r-uniform　hypergraph　such　that　each　pair　of　vertices　from　distinct　parts　is　contained　in　one　hyperedge　exactly.　An　immediate　corollary　of　this　result　is　that　exlinr　(n,　Kr++1)　=　nr2　for　sufficiently　large　n　and　r|　n,　2　where　exlinr　(n,　Kr++1)　is　the　maximum　number　of　edges　of　an　n-vertex　Kr++1-free　linear　r-uniform　hypergraph,　i.e.,　the　linear　Tur\＇an　number　of　Kr++1　©　2022　Society　for　Industrial　and　Applied　Mathematics