Sx [ ] = x must yield a

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1 Math -b Leture #5 Notes This wee we start with a remider about oordiates of a vetor relative to a basis for a subspae ad the importat speial ase where the subspae is all of R. This freedom to desribe vetors i R usig oordiates other tha the stadard oordiates the allows us to defie the matrix of a liear trasformatio relative to a differet basis. We ll the itrodue the idea of a vetor spae (or geeral liear spae) ad the eessary axioms that assure that virtually all of the defiitios ad ostrutios developed for R will also hold i these geeral liear spaes. Coordiates relative to a basis Defiitio: Suppose V R is a subspae (whih ould be all of R or ay proper subspae) ad let { v, v,, v} be a basis for V (hee dim( V) ). Ay vetor x V a be uiquely expressed as x v+ v + + v for some salars {,, } the basis { v v v }.,,,. The salars are alled the oordiates of x relative to Fidig oordiates of a vetor relative to a basis for a subspae: I terms of matries, we have x v v Sx [ ]. This says simply that the system of liear equatios Sx [ ] x must yield a x, ad we refer to this as the oordiate vetor for x relative to the basis. uique solutio [ ] Fidig oordiates of a vetor relative to a basis { v v v },,, for all of R : I this ase, the hage of basis matrix S is a matrix S v v. Its olums are liearly idepedet ad its ra is, so x S x it s ivertible. So we have [ ] x Sx ad [ ]. Matrix of a liear trasformatio relative to a alterate basis Defiitio: The matrix of a liear trasformatio T : matrix [ T] [ T( v )] [ T( v )] R R relative to the basis { v v v },,, is the. If the liear trasformatio T is represeted by the matrix A relative to the stadard basis { e e e }, we ofte simply write [ A] [ Av ] [ Av ],,,. Relatig matries of a liear trasformatio relative to differet bases If we mae use of the relatios x Sx [ ] ad [ x] S x together with the defiitio of a matrix for a liear trasformatio relative to a basis, we have the followig diagram to explai how these matries are related: A { R, } { R, } S S A { R, } { R, } revised February 7, 07

2 S AS We usually express this relatioship algebraially as either [ ] or [ ]. There s aother way to see this algebrai relatioship without a diagrammati roadmap. If we iterpret the olums of the hage of basis matrix S as well as the defiitio of [ A] we observe that: Se ( ) ( ) [ ] v ASe Av S ASe S Av S AS e Av S AS [ A] Se v ASe Av S ASe S ( Av ) S AS e Av A A SA S ( ) [ ] xample (part ): Suppose we have the basis {,, 3},, 3 v v v for R ad that a liear 3 T ( v) v trasformatio is defied i terms of how it ats o these basis vetors with T ( v) v3. If we deote the T ( v3) v + v matrix of this liear trasformatio relative to the stadard basis by A ad relative to the basis by [ A], fid the matrix A. Solutio: There is o alulatio eessary i determiig the matrix [ A]. This is atually extremely easy, perhaps we should eve say obvious. Observe that: T ( v ) v 0 0 T( ) 3 [ T ] [ T( ) ] [ T( ) ] [ T( 3) v v ] 0 v v v [ A] T ( v3) v + v 0 0 The simpliity of how this liear trasformatio is defied relative to the basis yields a orrespodigly simple matrix [ A] relative to this basis. To determie the matrix A relative to the stadard basis, we use. From the basis we have S ad we a alulate S 4 5 4, so A [ A] S AS ad solve for A SA [ ] S The moral is that if we hoose to wor oly with the stadard basis, ad if a liear trasformatio does ot at i a simple way relative to the stadard basis, the its stadard matrix will most liely ot be very simple. Fidig a basis relative to whih a give liear trasformatio ats simply will be a etral idea i the ourse i the omig wees. xample (part ): The basis i the above example is suh that the 3rd basis vetor is perpediular (orthogoal) to the first two vetors. Suppose V spa { v, v } is the plae spaed by the first two vetors. Fid the matrix for orthogoal projetio of ay vetor oto this plae. revised February 7, 07

3 T ( v) v 0 0 Solutio: Relative to the give basis, we observe that T ( v) v. So [ A] 0 0. As i the previous T ( v3) ase, we have A SA [ ] S This matrix will have ra beause its image is -dimesioal. Oe agai we see that the matrix is ompliated relative to the stadard basis but quite simple relative to a basis that is well-suited to the trasformatio. Geeral Liear Spaes (Vetor Spaes) Though we have dealt exlusively so far with R ad its subspaes, almost everythig that we have developed so far will wor the same way i ay spae where we a add elemets ad sale elemets i a maer aalogous to the way we add ad sale vetors. Defiitio: A vetor spae V is a set, with additio ad salig of elemet defied for all elemets of the set, that is losed uder additio ad salig, otais a zero elemet (0), ad satisfies the followig axioms: For all f, gh, Vad salars,, () ( f + g) + h f + ( g+ h) (5) ( f + g) f + g () f + g g+ f (6) ( + ) f f + f (3) f + 0 f (7) ( f) ( ) f (4) f + ( f) 0 (8) f f We ll deal primarily with the ase where the salars are real umbers. Suh a vetor spae is alled a real vetor spae. We ould also use omplex salars i whih ase we d all this a omplex vetor spae. Though we a give defiitios ad prove theorems about vetor spaes i geeral, it s helpful to develop a library of examples to whih we a refer.. R is a vetor spae. Ideed, the motivatio for our defiitio ad axioms is to defie vetor spaes to be spaes whih are fudametally lie R. All of the required axioms are familiar fats about vetors i R.. Ay subspae of R is a vetor spae. All of the axioms are iherited ad every subspae otais the zero vetor, ad the defiitio of subspae esures that a subspae is losed uder additio ad salig. 3. The omplex umbers { a + bi : a, b are real umbers, i } a be viewed as a real vetor spae where additio is defied by ( a+ bi ) + ( a + bi ) ( a+ a) + ( b+ b) i ad salig by a real umber is defied by ( a + bi) a + bi. Note that we do ot defie multipliatio of omplex umbers i this otext. The omplex umbers otai the zero elemet i ad all of the axioms follow from orrespodig fats about real umbers. m 4. M( m, ) { m matries with real etries} R is a real vetor spae with additio ad salig of matries defied etry-wise. The m zero matrix is the zero elemet ad the axioms are all ow properties of matrix algebra. Note that i this otext we do ot defie the produt of matries. 5. F(, ) F( ) { futios f : with domai } RR R R R R is a real vetor spae where additio of futios ad salig of futios are defied poitwise by ( f+ g)( x) f( x) + gx ( ) ad ( f )( x) f ( x). The zero elemet i this ase is the futio that is idetially zero for all x. (This is quite differet tha just the real umber 0.) Oe agai, the axioms all follow from familiar fats about real umbers. 3 revised February 7, 07

4 6. P { real polyomials of degree } { a0 + ax + + ax : a0, a,, a R} is a real vetor spae. Note that we must allow all polyomials less tha or equal to beause we might add two polyomials (or sale by 0) ad get a polyomial of lesser degree. 7. C 0 (, ) C 0 ( ) { otiuous futios f : } RR R R R is a real vetor spae. Closure follows from theorems of Calulus that the sum of otiuous futios is otiuous ad a salar multiple of a otiuous futio is also otiuous. The zero futio is learly otiuous, ad the axioms are all easily verified. Defiitio: A subspae W of a vetor spae V is a subset that is losed uder additio ad salig of elemets. That is, for ay vetors v, v W ad salars,, it must be the ase that v+ v W. We write W V. 8. C (, ) C ( ) { differetiable futios f : } RR R R R is a real vetor spae. Closure follows from the theorems of Calulus that ( f + g) f + g ad ( f ) f. The zero futio is learly differetiable. Note: All polyomials are differetiable, ad a (hopefully) familiar theorem of Calulus tells us that 0 differetiable futios must be otiuous, so for ay, P C ( R) C ( R) F( R ). 9. C (, ) C ( ) { futios f : that are at least times differetiable} RR R R R is a real vetor spae. This follows similarly from Calulus theorems. The zero futio is differetiable to all orders. 0. C (, ) C ( ) { futios f : that are differetiable to all orders} RR R R R is a real vetor spae. This also follows from the Calulus theorems above. The zero futio is differetiable to all orders. 0 Note: For ay, P0 P P P P C ( R) C ( R) C ( R) C ( R) F( R). It s also importat to ote that whe dealig with spaes of futios, these spaes are muh larger tha R. To better uderstad this, we ll eed some more defiitios, but some of the importat details will have to wait util you tae ourse i aalysis ad topology. May of the defiitios whe worig with vetors spaes are essetially the same as those i R. Defiitio: Give a olletio of elemets { f, f,, f} V spa { f,, f} { f+ + f where, are salars} Defiitio: A set of elemets { }, we defie the spa of these elemets as:. f, f,, f V is alled liearly idepedet if give ay liear ombiatio of the form f + + f 0, this implies that 0. That is, there is o otrivial way to ombie these vetors to yield the zero elemet. Defiitio: Give a subspae W V, a olletio of elemets { f, f,, f} W is alled a basis of W if Spa { f, f,, f} W ad { f, f,, f} are liearly idepedet. A basis is a miimal spaig set ad, as was the ase i R, if a basis for W osists of fiitely may elemets the ay other basis will have the same umber of elemets, the dimesio of W. It is importat to ote, however, that it will ofte be the ase, espeially i the ase of futio spaes, that a subspae might ot be spaed by fiitely may elemets. Coordiates relative to a basis Defiitio: If { f,, f} is a basis for a fiite dimesioal vetor spae V (or a subspae of V), ad if f V, the f a be expressed uiquely as f f + + f for salars {,, }. These uiquely 4 revised February 7, 07

5 determied salars are alled the oordiates of f relative to this basis. If we express these oordiates as a olum vetor (effetively a vetor i f. R ), we deote this oordiate vetor by [ ] xample: P { a0 + ax + + ax : a0, a,, a R} is a ( + )-dimesioal vetor spae with basis {, xx,,, x} argumet regardig liear idepedee is a little subtle. To show liear idepedee of {, xx,,, x}. The oeffiiets of a polyomial are the the oordiates relative to this (stadard) basis. The would set a0 + ax + + ax 0 ad try to the show that all the oeffiiets must be 0. It s importat to uderstad that this is a equatio i P, ot i the spae of real umbers, so we re ot seeig roots. The 0 o the right-had-side is ot the umber 0, but rather the futio that is idetially 0 for all x. Thus the equatio a0 + ax + + ax 0 for all x. I partiular, whe x 0 this gives a 0 0. You a reaso similarly (for example, by substitutig a variety of other values for x) that all the other oeffiiets must also be 0. If we loo speifially at P { a+ bx+ x : ab,, R }, the stadard basis would be {, xx, } also hoose to express these polyomials i terms of powers of ( x ), i.e. {, x, ( x ) }, we but we might. Give a a polyomial of the form px ( ) a+ bx ( ) + x ( ), we would say that [ p] b ad we a multiply out to get p( x) a + b( x ) + ( x ) a + bx b + x 4x + 4 ( a b + 4 ) + ( b 4 ) x + x. So a b+ 4 4 a [ p] b 4 0 4b S[ p] 0 0 hage of basis matrix ad the relatio [ p] S[ p]. We would the, quite aturally, refer to this matrix S as a is remiiset of the way we haged oordiates i R3. We a also do this the other way aroud by startig with a polyomial of the form stadard oordiates [ p] we might write: a b p( x) a bx x + + with., ad the figure out its oordiates relative to the basis {, x, ( x ) } p( x) a+ bx+ x a+ b( x + ) + ( x + ) a+ b( x ) + b+ ( x ) + 4 ( x ) + 4 ( a+ b+ 4 ) + ( b+ 4 )( x ) + ( x ) So [ p] b b S [ p] a+ b+ 4 4 a 4 4. You a verify that S S I Defiitio: Give two vetors spaes V ad W, a futio T : V W is alled a liear trasformatio if for all elemets f, f V ad for all salars,, T satisfies T( f+ f) T ( f) + T( f). This a also be expressed by sayig that T preserves additio ad salar multipliatio. We all the iput spae V the domai of T ad we all the output spae W the odomai. Defiitio: Suppose T : V W is a liear trasformatio. We defie: image( T ) im( T ) T ( f ): f V W erel( T) er( T) f V : T( f) 0 V { } ad { } 5 revised February 7, 07

6 These are both subspaes. The argumet is the same as we ve see before. Matrix of a liear trasformatio relative to bases for the domai ad odomai Defiitio: If V is a fiite-dimesioal vetor spae with basis { f,, f} ad W is a fiite-dimesioal C,, m, ad if T : V vetor spae with basis { g g } W is a liear trasformatio, we defie the matrix of T relative to these bases as: [ T] [ T( f) ] [ T( f ) ]. This will be a m matrix.,c C C This is simpler i the ase where the domai ad odomai oiide. I that ase we might use the same basis for both ad defie [ T] [ T( f) ] [ T( f ) ]. I this ase, [ T ] would be a (square) matrix. Note: It is sometimes desirable to fid the erel ad image of a liear trasformatio by hoosig a basis or bases to express everythig i oordiates ad the fidig the erel ad image of the matrix for the liear trasformatio relative to this basis or bases. Defiitio: Give a liear trasformatio T : V W, we defie: ra( T) dim(im T) ad ullity( T) dim(er T) whe these subspaes have fiite dimesio. We a also state (without proof) the orrespodig fat regardig the relatioship betwee ra ad ullity. Ra-Nullity Theorem: If T : V ra( T) + ullity( T) dim( V). W is a liear trasformatio ad V has fiite dimesio, the Defiitio: A liear trasformatio T : V W is alled a isomorphism if it is oe-to-oe ad oto its odomai. That is, for every g W there is a uique f V suh that T( f) g. For example, the orrespodee betwee a polyomial a [ p] b i p( x) a bx x + + i 3 R is learly liear, oe-to-oe, ad oto; so we a say that P ad P ad its oordiates 3 R are isomorphi. xample: I ay spae osistig of differetiable futios, the differetiatio operator D( f) f is a liear trasformatio. This follows from the Calulus fats that D( f + g) ( f + g) f + g D( f) + D( g) ad D( f ) ( f ) f D( f ). If we restrit our attetio to a fiite dimesioal spae suh as P, we ote that D D x, so relative to the basis {, xx, }, we have the matrix [ D] 0 0. The erel of this D x x matrix is spa 0 { ostat futios}, ad the image is spa 0, { a + bx : a, b R } P. Said differetly, the set of all suh futios whose derivative is 0 osists of the ostat futios, ad if you differetiate a quadrati futio you ll get a polyomial of degree. It should be lear that this liear trasformatio is defiitely NOT a isomorphism. 6 revised February 7, 07

7 xample: Let V R ad let T ( A) A for ay matrix A. It s easy to see that this is liear. Fid bases for its erel ad its image ad the matrix of T relative to the basis,,, ,,, is a basis.) Is T a isomorphism? (It s easy to see that { } 3 4 a b a b a+ b ( a+ b) 0 0 Solutio: Let A d ad alulate T ( A ) d + d ( + d). If T ( A ) 0 0, the 0 0 a+ b 0 ad + d 0, so b a ad d. So if A er( T ), the A a a a So, 0 0 is a basis for er( T ) sie these are learly liearly idepedet. Therefore ullity( T ). a+ b ( a+ b) 0 0 Aythig i the image is of the form T( A ) ( a+ b) + ( + d) + d ( + d) 0 0, so 0 0, 0 0 is a basis for im( T ) ad ra( T ). Note that ra( ) ullity( ) 4 dim( ) T + T V. er( T ) 0 ). There are a ouple of good ways to fid the matrix of T. This is NOT a isomorphism (sie { } y the olums: We ould alulate what the liear trasformatio does to eah of the basis elemets i order to determie the olums of the matrix. This is a bit tedious, but it goes lie this: y the rows: If the iput elemet is a b a+ b ( a+ b) T ( ) d + d ( + d) 0 T( ) T( ) T( 3) T( 4) a b d A, its oordiates are [ ] A, ad its oordiates are [ ] 0 0 a 0 0b d Sie [ T( A) ] [ T] [ A], it follows that [ T ] [ T ] a b A. The output elemet is d a+ b 0 0 a a+ b 0 0b T ( A). + d d 0 0 d revised February 7, 07

8 It is worth otig that the ra of this matrix is, ad we ould alteratively have used this matrix to determie the erel ad image of the liear trasformatio that it represets. Note: A liear trasformatio T : V W where V ad W are fiite-dimesioal will be a isomorphism if ad oly if its matrix relative to suitable bases for V ad W is a ivertible matrix. Notes by Robert Witers 8 revised February 7, 07