שינויים
/* qn */
Prove that if dim(U+span{(v+w)})<dim(U+sp{v}) then v,w∉U
'''Assume v,w∈U then U+sp{v+w}=U=U+sp{v}. Therefore dim(U+sp{v+w})=dim(U)=dim(U+sp{v}). contradiction
so v∉U or w∉U.
'''If (w∉U and v∈U) then v+w∉U and we get
<math>dim(U+span{(v+w)})=dim(U)+dim(sp{v+w})-dim(U\bigcap sp{v+w})=dim(U)+1-0\geq dim(U)=dim(U+sp{v})</math>. contradiction
'''If (v∉U and w∈U) then v+w∉U and we get
<math>dim(U+span{(v+w)})=dim(U)+dim(sp{v+w})-dim(U\bigcap sp{v+w})=dim(U)+1-0= dim(U)+dim(sp{v})-dim(U\bigcap sp{v})=dim(U+sp{v})</math>. contradiction
Adi
