Transcription of Linear Algebra 2: Direct sums of vector spaces
1 Linear Algebra 2: Direct sums of vector spacesThursday 3 November 2005 Lectures for Part A of Oxford FHS in Mathematics and Joint Schools Direct sums of vector spaces Projection operators Idempotent transformations Two theorems Direct sums and partitions of the identityImportant note:Throughout this lectureFis a field andVis a vector space sum decompositions, IDefinition:LetU,Wbe subspaces ofV. ThenVis said tobe thedirect sumofUandW, and we writeV=U W,ifV=U+WandU W={0}.Lemma:LetU,Wbe subspaces ofV. ThenV=U Wifand only if for everyv Vthere exist unique vectorsu Uandw Wsuch thatv=u+ operatorsSuppose thatV=U W. DefineP:V Vas follows. Forv Vwritev=u+wwhereu Uandw W: then defineP(v) := :(1)Pis well-defined;(2)Pis Linear ;(3)ImP=U, KerP=W;(4)P2= :Pis called on projection operatorsNote thatVis finite-dimensional.
2 Choose a basisu1, .. , urforUand a basisw1, .. , wmforU. Then the matrixofPwith respect to the basisu1, .. , ur, w1, .. , wmofVis(Ir00 0).Note the projection ontoUalongWthenI Pis theprojection operators: a theoremTerminology:An operatorTsuch thatT2=Tis said to idempotent operator is a projection theorem about :V Vbe the projection :V Vbe a Linear transformation. ThenP T=T Pif andonly ifUandWareT-invariant(that isT U6 UandT W6W). sum decompositions, IIDefinition:Vis said to bedirect sumof subspacesU1,..,Uk,and we writeV=U1 Uk,if for everyv Vthere existuniquevectorsui Uifor 16i6ksuch thatv=u1+ + :U1 U2 Uk= ( ((U1 U2) U3) Uk).Note:IfUi6 Vfor 16i6kthenV=U1 Ukif and onlyifV=U1+U2+ +UkandUr i6=rUi={0}for isNOTsufficient thatUi Uj={0}wheneveri6= :IfV=U1 U2 UkandBiis a basis ofUithenB1 B2 Bkis a basis ofV.
3 In particular, dimV=k i= of the identityLetP1, .. , Pkbe Linear mappingsV Vsuch thatP2i=PiforalliandPiPj= 0 wheneveri6=j. IfP1+ +Pk=Ithen{P1, .. , Pk}is known as apartition of the identity :IfPis a projection then{P, I P}is a partition ofthe thatV=U1 Uk. LetPibe the projec-tion ofVontoUialong j6=iUj. Then{P1, .. , Pk}is a partitionof the identity , if{P1, .. , Pk}is a partitionof the identity onVandUi:= ImPithenV=U1
