Chapter 5 Linear Transformations and Operators

1 Chapter 5 Linear Transformations The Algebra of Linear TransformationsTheorem vector spaces over the fieldF. LetTandUbetwo Linear Transformations fromVintoW. The function(T+U)defined pointwiseby(T+U) (v) =Tv+Uvis a Linear transformation fromVintoW. Furthermore, ifs F, the function(sT)defined by(sT) (v) =s(Tv)is also a Linear transformation fromVintoW. The set of all Linear transformationfromVintoW, together with the addition and scalar multiplication defined above,is a vector space over the thatTandUare Linear transformation fromVintoW. For(T+U)defined above, we have(T+U) (sv+w) =T(sv+w) +U(sv+w)=s(Tv) +Tw+s(Uv) +Uw=s(Tv+Uv) + (Tw+Uw)=s(T+U)v+ (T+U)w,8384 Chapter 5.

2 Linear Transformations AND Operators which shows that(T+U)is a Linear transformation . Similarly, we have(rT) (sv+w) =r(T(sv+w))=r(s(Tv) + (Tw))=rs(Tv) +r(Tw)=s(r(Tv)) +rT(w)=s((rT)v) + (rT)wwhich shows that(rT)is a Linear verify that the set of Linear Transformations fromVintoWtogether with theoperations defined above is a vector space, one must directly check the conditionsof Definition These are straightforward to verify, and we leave this exerciseto the denote the space of Linear Transformations fromVintoWbyL(V,W). NotethatL(V,W)is defined only whenVandWare vector spaces over the same ann-dimensional vector space over the fieldF, and letWbe anm-dimensional vector space overF.

3 Then the spaceL(V,W)is finite-dimensional and has ,W, andZbe vector spaces over a fieldF. LetT L(V,W)andU L(W,Z). Then the composed functionUTdefined by(UT) (v) =U(T(v))is a Linear transformation ,v2 Vands F. Then, we have(UT) (sv1+v2) =U(T(sv1+v2))=U(sTv1+Tv2)=sU(Tv1) +U(Tv2)=s(UT) (v1) + (UT) (v2),as a vector space over the fieldF, alinear operatoronVisa Linear transformation THE ALGEBRA OF Linear TRANSFORMATIONS85 Definition Linear transformationTfromVintoWis calledinvertibleifthere exists a functionUfromWtoVsuch thatUTis the identity function onVandTUis the identity function onW. IfTis invertible, the functionUis uniqueand is denoted byT 1.

4 Furthermore,Tis invertible if and only one-to-one:Tv1=Tv2= v1= onto: the range the vector spaceVof semi-infinite real sequencesR wherev= (v1,v2,v3,..) Vwithvn Rforn N. LetL:V Vbe theleft-shift Linear transformation defined byLv= (v2,v3,v4,..)andR:V Vbe the right-shift Linear transformation defined byRv= (0,v1,v2,..).Notice thatLis onto but not one-to-one andRis one-to-one but not onto. Therefore,neither transformation is the normed vector spaceVof semi-infinite real sequencesR with the standard Schauder basis{e1,e2,..}. LetT:V Vbe the lineartransformation that satisfiesTei=i 1eifori= 1,2,.. Let the Linear transfor-mationU:V VsatisfyUei=ieifori= 1,2.

5 It is easy to verify thatU=T 1andUT=TU= example should actually bother you somewhat. SinceTreduces vectorcomponents arbitrarily, its inverse must enlarge them arbitrarily. Clearly, this is nota desirable property. Later, we will introduce a norm for Linear transforms whichquantifies this vector spaces over the fieldFand letTbe alinear transformation fromVintoW. IfTis invertible, then the inverse functionT 1is a Linear transformation vectors inWand lets F. Definevj=T 1wj, forj= 1,2. SinceTis a Linear transformation , we haveT(sv1+v2) =sT(v1) +T(v2) =sw1+ 5. Linear Transformations AND OPERATORSThat is,sv1+v2is the unique vector inVthat maps tosw1+w2underT.

6 It followsthatT 1(sw1+w2) =sv1+v2=s(T 1w1)+T 1w2andT 1is a Linear a mapping between algebraic structures which preservesall relevant structure. Anisomorphismis a homomorphism which is also invert-ible. For vector spaces, the relevant structure is given by vector addition and scalarmultiplication. Since a Linear transformation preserves both of these operation, it isalso avector space homomorphism. Likewise, an invertible Linear transformation isavector space Linear Functionals on Vector SpacesDefinition a vector space over a fieldF. A Linear transformationffromVinto the scalar fieldFis called alinear is,fis a functional onVsuch thatf(sv1+v2) =sf(v1) +f(v2)for allv1,v2 Vands a field and lets1.

7 ,snbe scalars inF. Then the func-tionalfonFndefined byf(v1,..,vn) =s1v1+ +snvnis a Linear functional. It is the Linear functional which is represented by the matrix[s1s2 sn]relative to the standard ordered basis forFn. Every Linear functional onFnis ofthis form, for some scalarss1,.., a positive integer andFa field. IfAis ann nmatrixwith entries inF, thetraceofAis the scalartr(A) =A11+A22+ + Linear FUNCTIONALS ON VECTOR SPACES87 Example trace function is a Linear functional on the matrix spaceFn nsincetr(sA+B) =n i=1(sAii+Bii)=sn i=1 Aii+n i=1 Bii=str(A) + tr(B).Example [a,b]be a closed interval on the real line and letC([a,b])bethe space of continuous real-valued functions on[a,b].

8 ThenL(g) = bag(t)dtdefines a Linear functionalLonC([a,b]).Definition a vector space. The collection of all Linear functionalsonV, denotedL(V,F), forms a vector space. We also denote this space byV andcall it thedual following theorem shows that, ifVis finite dimensional, thendimV = this case, one actually finds thatVis isomorphic toV . Therefore, the two spacescan be identified with each other so thatV=V for finite a finite-dimensional vector space over the fieldF, andletB=v1,..,vnbe a basis forV. There is a unique dual basisB =f1,..,fnforV such thatfj(vi) = ij. For each Linear functional onV, we havef=n i=1f(vi)fiand for each vectorvinV, we havev=n i=1fi(v) 5.

9 Linear Transformations AND ,..,vnbe a basis forV. According to Theorem , there is aunique Linear functionalfionVsuch thatfi(vj)= , we obtain fromBa set ofndistinct Linear functionalsf1,..,fnonV. Thesefunctionals are linearly independent; suppose thatf=n i=1sifi,thenf(vj)=n i=1sifi(vj)=n i=1si ij= particular, iffis the zero functional,f(vj)= 0forj= 1,..,nand hence thescalars{sj}must all equal0. It follows that the functionalsf1,..,fnare linearlyindependent. SincedimV =n, we conclude thatB =f1,..,fnforms a basisforV , thedual , we want to show that there is a unique basis which is dual toB. Iffis alinear functional onV, thenfis some Linear combination off1.

10 ,fnwithf=n i= , by construction, we must havesj=f(vj)forj= 1,..,n. Simi-larly, ifv=n i= a vector inV, thenfj(v) =n i=1tifj(vi) =n i=1ti ij= is, the unique expression forvas a Linear combination ofv1,..,vnisv=n i=1fi(v) OPERATOR NORMS89 One important use of the dual space is to define the transpose of a Linear trans-form in a way that generalizes to infinite dimensional vector spaces. LetV,Wbe vector spaces overFandT:V Wbe a Linear transform. Ifg W is a Linear functional on W ( ,g:W F), theng(Tv) V is a linearfunctional onV. ThetransposeofTis the mappingU:W V defined byf(v) =g(Tv) V for allg W . IfV,Ware finite-dimensional, then one canidentifyV=V andW=W via isomorphism and recover the standard transposemappingU:W Vimplied by the matrix details of this definition are not used in the remainder of these notes, but canbe useful in understanding the subtleties of infinite dimensional spaces.