Chapter 3. Vector Spaces

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1 3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce to nother. We ll see prllel behvior between these liner trnsformtions nd the mtrix trnsformtions of Section 2.3. In fct, we use ordered bses to ssocite mtrices with liner trnsformtions between generl finite dimensionl vector spces. Definition 3.9. A function T tht mps vector spce V into vector spce V is liner trnsformtion if it stisfies: (1) ( u + v) = T( u) + T( v), nd (2) T(r u) = rt( u), for ll vectors u, v V nd for ll sclrs r R. Note 3.4.A. Exercise clims tht the condition of T : V V being liner is equivlent to the condition T(r u + s v) = rt( u) + st( v) for ll u, v V nd for ll r,s R. Notice tht the sme clim ws estblished for V nd V Eucliden spces in Exercises We cn conclude (from Mthemticl Induction, see Appendix A) tht for v 1, v 2,..., v n V nd r 1,r 2,...,r n R, we hve T(r 1 v 1 + r 2 v r n v n ) = r 1 T( v 1 ) + r 2 T( v 2 ) + + r n T( v n ). Exmple 3.4.A. Let F be the vector spce of ll functions mpping R into R (see Exmple 3.1.3). Let be nonzero sclr nd define T : F F s T(f) = f. Is T liner trnsformtion?

2 3.4 Liner Trnsformtions 2 Definition. For liner trnsformtion T : V V, the set V is the domin of T nd the set V is the codomin of T. If W is subset of V, then T[W] = {T( w) w W } is the imge of W under T. T[V ] is the rnge of T. For W V, T 1 [W ] = { v V T( v) W } is the inverse imge of W under T. T 1 [{ 0 }] if the kernel of T, denoted ker(t). Notice tht ker(t) = { v V T( v) = 0 }. Exmple 3.4.B. Let F be the vector spce of ll functions mpping R into R (see Exmple 3.1.3). Let be nonzero sclr nd define T : F F s T(f) = f, s in Exmple 3.4.A. Describe the kernel of T. Definition. Let V,V nd V be vector spces nd let T : V V nd T : V V be liner trnsformtions. The composition trnsformtion T T : V V is defined by (T T)( v) = T (T( v)) for v V. Exmple. Pge 214 Exmple 1. Let F be the vector spce of ll functions f : R R (see Exmple 3.1.3), nd let D be its subspce of ll differentible functions. Show tht differentition is liner trnsformtion of D into F. Exmple. Pge 215 Exmple 3. Let C,b be the set of ll continuous functions mpping [,b] R. Then C,b is vector spce (bsed on n rgument similr to tht which justifies tht C = {f F f is continuous} is subspce of F, s mentioned in Note 3.2.A). Prove tht T : C,b R defined by T(f) = b f(x) dx is liner trnsformtion. Such trnsformtion which mps functions to rel numbers is clled liner functionl.

3 3.4 Liner Trnsformtions 3 Exmple. Pge 215 Exmple 4. Let C be the vector spce of ll continuous functions mpping R into R (see Note 3.2.A). Let R nd let T : C C be defined by T (f) = x f(t) dt. Prove tht T is liner trnsformtion. Note. One might think tht the differentition opertor D : D F nd the opertor T : C C in the previous exmple re inverses of ech other (we hve not yet defined the inverse of liner trnsformtion from one generl vector spce to nother). This is not the cse, though, since T (f) = x f(t) dt implies tht ( x T (f)() = f(t) dt) = f(t) dt = 0, so T mps continuous functions to continuous functions which re 0 t x =. Now ech T (f) is differentible since d dx [T [ (f)] = d x dx f(t) dt] = f(x) by the Fundmentl Theorem of Clculus. If we define D = {f D d() = 0} (D is subspce of D bsed on n rgument similr to tht given in Exercise 3.2.4) then x= we hve T : C D, D : D C, nd for f C, ( x ) (D T )(f) = D(T (f)) = D f(t) dt = d [ x dx ] f(t) dt = f(x) = f. If f D (so f() = 0) AND f is continuously differentible (tht is, f is continuous) then (T D)(f) = T (D(f)) = T (f ) = x f (t) dt = f(x) f() = f(x) = f. So if we define D 1, = {f D f is continuous} ( subspce of D ), then we do hve tht the differentition D : D 1, C nd T : C D 1, re inverses of ech other.

4 3.4 Liner Trnsformtions 4 Note. Frleigh nd Beuregrd lso give n exmple of liner functionl T : F R defined for given c R s T(f) = f(c). This is n exmple of n evlution functionl (see Exmple 3.4.2). In Exmple 3.4.5, the uthors show tht T : D D defined, for 0, 1,..., n R, s T(f) = n f (n) (x) + n 1 f (n 1) (x) f (x) + 0 f(x) is liner trnsformtion. This exmple plys fundmentl role in the study of nth-order liner differentil equtions with constnt coefficients (where the tools developed for mtrices re useful). Theorem 3.5. Preservtion of Zero nd Subtrction Let V nd V be vectors spces, nd let T : V V be liner trnsformtion. Then (1) T( 0) = 0, nd (2) T( v 1 v 2 ) = T( v 1 ) T( v 2 ), for ny vectors v 1 nd v 2 in V. Theorem 3.6. Bses nd Liner Trnsformtions. Let T : V V be liner trnsformtion, nd let B be bsis for V. For ny vector v in V, the vector T( v) is uniquely determined by the vectors T( b) for ll b B. In other words, if two liner trnsformtions hve the sme vlue t ech bsis vector b B, then the two trnsformtions hve the sme vlue t ech vector in V.

5 3.4 Liner Trnsformtions 5 Theorem 3.7. Preservtion of Subspces. Let V nd V be vector spces, nd let T : V V be liner trnsformtion. (1) If W is subspce of V, then T[W] is subspce of V. (2) If W is subspce of V, then T 1 [W ] is subspce of V. Theorem 3.4.A. (Pge 229 number 46) Let T : V V be liner trnsformtion nd let T( p) = b for prticulr vector p in V. The solution set of T( x) = b is the set { p + h h ker(t)}. Definition. A trnsformtion T : V V is one-to-one if T( v 1 ) = T( v 2 ) implies tht v 1 = v 2 (or by the contrpositive, v 1 v 2 implies T( v 1 ) T( v 2 )). Trnsformtion T is onto if for ll v V there is v V such tht T( v) = v. Corollry 3.4.A. One-to-One nd Kernel. A liner trnsformtion T is one-to-one if nd only if ker(t) = { 0}. Definition Let V nd V be vector spces. A liner trnsformtion T : V V is invertible if there exists liner trnsformtion T 1 : V V such tht T 1 T is the identity trnsformtion on V nd T T 1 is the identity trnsformtion on V. Such T 1 is clled n inverse trnsformtion of T. Theorem 3.8. A liner trnsformtion T : V V is invertible if nd only if it is one-to-one nd onto V. When T 1 exists, it is liner.

6 3.4 Liner Trnsformtions 6 Exmple 3.4.C. Let F be the vector spce of ll functions mpping R into R (see Exmple 3.1.3). Let be nonzero sclr nd define T : F F s T(f) = f, s in Exmple 3.4.A. Determine if T is invertible. If so, find its inverse. Note. It is t this stge tht Frleigh nd Beuregrd introduce the Fundmentl Theorem of Finite Dimensionl Vector spces (see Theorem 3.3.A). They define n isomorphism s one-to-one nd onto liner trnsformtion (s we did in Section 3.3, though we didn t use the lnguge of liner trnsformtion t tht time). Their comments on isomorphisms on pge 221 re certinly worth reding. For completeness, we now stte their version of the Fundmentl Theorem of Finite Dimensionl Vector Spces long with the nme they give it. Theorem 3.9. Coordintiztion of Finite-Dimensionl Spces. Let V be finite-dimensionl vector spce with ordered bsis B = ( b 1, b 2,..., b n ). The mp T : V R n defined by T( v) = v B, the coordinte vector of v reltive to B, is n isomorphism. Tht is, ny n-dimensionl vector spce is isomorphic to R n. Note. Just s mtrices represented liner trnsformtions mpping R n R m (see Corollry 2.3.A, Stndrd Mtrix Representtion of Liner Trnsformtions ), we cn use the coordintiztion of generl finite dimensionl vector spces V nd V to represent liner trnsformtion mpping V V with mtrix.

7 3.4 Liner Trnsformtions 7 Theorem Mtrix Representtions of Liner Trnsformtions. Let V nd V be finite-dimensionl vector spces nd let B = ( b 1, b 2,..., b n ) nd B = ( b 1, b 2,..., b m) be ordered bses for V nd V, respectively. Let T : V V be liner trnsformtion, nd let T : R n R m be the liner trnsformtion such tht for ech v V, we hve T( v B ) = T( v) B. Then the stndrd mtrix representtion of T is the mtrix A whose jth column vector is T( b j ) B, nd T( v) B = A v B for ll vectors v V. Definition The mtrix A of Theorem 3.10 is the mtrix representtion of T reltive to B,B. Exmples. Pge 227 Number 18, Pge 227 Number 22, Pge 227 Number 24. Note. Let T : V V where B is bsis for V nd B is bsis for V. By Theorem 3.8, T 1 is liner when it exists. So it hs mtrix representtion reltive the B,B. The next result gives this mtrix representtion in terms of the mtrix representtion of T reltive to B,B. Theorem 3.4.B. The mtrix representtion of T 1 reltive to B,B is the inverse of the mtrix representtion of T reltive to B,B. Exmples. Pge 228 Number 28, Pge 229 Number 44, Pge 226 Number 12. Revised: 10/24/2018