Linear Transformations

Transcription

1 Liear rasformatios 6. Itroductio to Liear rasformatios 6. he Kerel ad Rage of a Liear rasformatio 6. Matrices for Liear rasformatios 6.4 rasitio Matrices ad Similarity 6.5 Applicatios of Liear rasformatios 6.

2 6. Itroductio to Liear rasformatios A fuctio that maps a ector space V ito a ector space W: V W V W mappig :,, : ector spaces V: the domai of W: the codomai of Image of uder : If is a ector i V ad w is a ector i W such that w, the w is called the image of uder For each, there is oly oe w he rage of : he set of all images of ectors i V see the figure o the et slide 6.

3 he preimage of w: he set of all i V such that =w For each w, may ot be uique he graphical represetatios of the domai, codomai, ad rage For eample, V is R, W is R, ad is the orthogoal projectio of ay ector, y, z oto the y-plae, i.e., y, z =, y, we will use the aboe eample may times to eplai abstract otios he the domai is R, the codomai is R, ad the rage is y-plae a subspace of the codomia R,, is the image of,, he preimage of,, is,, s, where s is ay real umber 6.

4 E : A fuctio from R ito R : R R,,, R a Fid the image of =-, b Fid the preimage of w=-, Sol: a,,,, b w,,,,, 4 hus {, 4} is the preimage of w=-, 6.4

5 Liear rasformatio : V, W: ector spaces : V W: A liear trasformatio of V ito W followig two propertiesare true if the u u, u, V cu c u, c R 6.5

6 Notes: A liear trasformatio is said to be operatio preserig because the same result occurs whether the operatios of additio ad scalar multiplicatio are performed before or after the liear trasformatio is applied u u cu c u Additio i V Additio i W Scalar multiplicatio i V Scalar multiplicatio i W A liear trasformatio : V V from a ector space ito itself is called a liear operator 6.6

7 6.7 E : Verifyig a liear trasformatio from R ito R Pf:,, :ay real umber, :ector i,,, c R u u u Vector additio :,,, u u u u u,,,,, u u u u u u u u u u u u u u u u

8 6.8,, Scalar multiplicatio cu cu u u c c u,,, u u c u u u u c cu cu cu cu cu cu c herefore, is a liear trasformatio

9 E : Fuctios that are ot liear trasformatios a f si si si si si si si b f c f f f f f f f I fact, f c cf f = si is ot a liear trasformatio f = is ot a liear trasformatio f = + is ot a liear trasformatio, although it is a liear fuctio 6.9

10 Notes: wo uses of the term liear. f is called a liear fuctio because its graph is a lie f is ot a liear trasformatio from a ector space R ito R because it preseres either ector additio or scalar multiplicatio 6.

11 6. Zero trasformatio: V W V u,, :, V Idetity trasformatio: V V : V, heorem 6.: Properties of liear trasformatios W V : u u the, 4 If c c c c c c c c c c = c for c=- u+-=u+- ad property c = c for c= Iteratiely usig u+=u+ ad c = c

12 E 4: Liear trasformatios ad bases Let : R R,,,,4,,,5,,,,, be a liear trasformatio such that Fid,, - Sol:,,,,,,,, Accordig to the fourth property o the preious slide that c c c c c c,,,,,,,,,,4,5,,, 7,7, 6.

13 E 5: A liear trasformatio defied by a matri he fuctio Sol: : R R is defied as a Fid, where, A b Show that is a liear trasformatio form R ito R a, R ector R ector A, 6,, 6 b u A u Au A u cu A cu c Au c u ector additio scalar multiplicatio 6.

14 6.4 heorem 6.: he liear trasformatio defied by a matri Let A be a m matri. he fuctio defied by A is a liear trasformatio from R ito R m Note: m m m m m m a a a a a a a a a a a a a a a a a a A A m R R : ector R ector m R If ca represeted by A, the is a liear trasformatio If the size of A is m, the the domai of is R ad the codomai of is R m

15 E 7: Rotatio i the plae Show that the L.. : R R gie by the matri cos A si si cos has the property that it rotates eery ector i R couterclockwise about the origi through the agle Sol:, y r cos, rsi r:the legth of :the agle from the positie -ais couterclockwise to the ector Polar coordiates: for eery poit o the yplae, it ca be represeted by a set of r, α y 6.5

16 A cos si cos si cos y si r cos cos r si si r si cos r cos si r cos r si si r cos cos r si r:remai the same, that meas the legth of equals the legth of +:the agle from the positie -ais couterclockwise to the ector hus, is the ector that results from rotatig the ector couterclockwise through the agle accordig to the additio formula of trigoometric idetities 6.6

17 E 8: A projectio i R he liear trasformatio A is called a projectio i R : R R is gie by If is, y, z, A y y z I other words, maps eery ector i R to its orthogoal projectio i the y-plae, as show i the right figure 6.7

18 Keywords i Sectio 6.: fuctio domai codomai image of uder rage of preimage of w liear trasformatio liear operator zero trasformatio idetity trasformatio 6.8

19 6. he Kerel ad Rage of a Liear rasformatio Kerel of a liear trasformatio : Let : V W be a liear trasformatio. he the set of all ectors i V that satisfy is called the kerel of ad is deoted by ker ker {, V} For eample, V is R, W is R, ad is the orthogoal projectio of ay ector, y, z oto the y-plae, i.e., y, z =, y, he the kerel of is the set cosistig of,, s, where s is a real umber, i.e. ker {,, s s is a real umber} 6.9

20 E : he kerel of the zero ad idetity trasformatios a If = the zero trasformatio : V W, the ker V b If = the idetity trasformatio : V V, the ker { } 6.

21 E 5: Fidig the kerel of a liear trasformatio A : R R ker? Sol: ker {,,,,,, ad,, R },,, 6.

22 G.-J. E. t t t t ker { t,, t is a real umber} spa{,,} 6.

23 heorem 6.: he kerel is a subspace of V Pf: he kerel of a liear trasformatio subspace of the domai V Let u ad by heorem 6. : V W is a ker is a oempty subset of V be ectorsi the kerel of.he u u cu c u c uker, ker u ker is a liear trasformatio uker cu ker hus, ker is a subspace of V accordig to heorem 4.5 that a oempty subset of V is a subspace of V if it is closed uder ector additio ad scalar multiplicatio 6.

24 6.4 E 6: Fidig a basis for the kerel 8 ad is i where, be defied by : Let A R A R R Fid a basis for ker as a subspace of R 5 Sol: o fid ker meas to fid all satisfyig = A =. hus we eed to form the augmeted matri first A

25 A G.-J. E. 4 8 s t s t s t s s t 4 4t 4 5 t B,,,,,,,, 4, : oe basis for the kerel of 6.5

26 Corollary to heorem 6.: Let : R he the R m kerel of be the liear frasformatio gie by is equal to the solutio space of m A a liear trasformatio : R R A ker NS A A, R subspace of R he kerel of equals the ullspace of A which is defied i heorem 4.6 o p.9 ad these two are both subspaces of R So, the kerel of is sometimes called the ullspace of A. 6.6

27 Rage of a liear trasformatio : Let : V W be a liear trasformatio. he the set of all ectors w i W that are images of ay ectors i V is called the rage of ad is deoted by rage rage { V} For the orthogoal projectio of ay ector, y, z oto the yplae, i.e., y, z =, y, he domai is V=R, the codomai is W=R, ad the rage is y-plae a subspace of the codomia R Sice,, s =,, =, the kerel of is the set cosistig of,, s, where s is a real umber 6.7

28 heorem 6.4: he rage of is a subspace of W he rage of a liear trasformatio : V W is a subspace of W Pf: heorem 6. rage is a oempty subset of W Sice u ad are ectorsi rage, ad we u u rage c u cu rage hae because uv Rage of is closed uder ector additio because u,, u rage is a liear trasformatio Rage of is closed uder scalar multi- plicatio because u ad cu rage because cuv hus, rage is a subspace of W accordig to heorem 4.5 that a oempty subset of W is a subspace of W if it is closed uder ector additio ad scalar multiplicatio 6.8

29 Notes: : V W is a liear trasformatio ker is subspace of V heorem 6. rage is subspace of W heorem

30 Corollary to heorem 6.4: Let he : R rage of R m be the liear trasformatio gie by is equal to the colum space of A. A, i.e. rage CS A Accordig to the defiitio of the rage of = A, we kow that the rage of cosists of all ectors b satisfyig A=b, which is equialet to fid all ectors b such that the system A=b is cosistet A=b ca be rewritte as a a a a a a A b a a a m m m herefore, the system A=b is cosistet iff we ca fid,,, such that b is a liear combiatio of the colum ectors of A, i.e. bcs A hus, we ca coclude that the rage cosists of all ectors b, which is a liear combiatio of the colum ectors of A or said bcs A. So, the colum space of the matri A is the same as the rage of, i.e. rage = CSA 6.

31 Use our eample to illustrate the corollary to heorem 6.4: For the orthogoal projectio of ay ector, y, z oto the yplae, i.e., y, z =, y, Accordig to the aboe aalysis, we already kew that the rage of is the y-plae, i.e. rage={, y, ad y are real umbers} ca be defied by a matri A as follows A, such that y y z he colum space of A is as follows, which is just the y-plae CS A, where, R 6.

32 E 7: Fidig a basis for the rage of a liear trasformatio Let : R R be defied by A, where is R ad Sol: A 8 Fid a basis for the rage of Sice rage = CSA, fidig a basis for the rage of is equialet to fiig a basis for the colum space of A 6.

33 G.-J. E. A B 4 8 c c c w w w w c 4 c5 4 w5 w, w, ad w are idepdet, so w, w, w ca 4 4 form a basis for CS B Row operatios will ot affect the depedecy amog colums c, c, ad c are idepdet, ad thus c, c, c is 4 4 a basis for CS A hat is,,, for the rage of,,,,,,,,, is a basis 6.

34 Rak of a liear trasformatio :V W : rak the dimesio of the rage of dimrage Accordig to the corollary to hm. 6.4, rage = CSA, so dimrage = dimcsa Nullity of a liear trasformatio :V W : ullity the dimesio of the kerel of dimker Accordig to the corollary to hm. 6., ker = NSA, so dimker = dimnsa Note: m If : R R is a liear trasformatio gie by A, the rak dimrage dim CS A rak A ullity dimker dim NS A ullity A he dimesio of the row or colum space of a matri A is called the rak of A he dimesio of the ullspace of A NS A { A} is called the ullity of A 6.4

35 heorem 6.5: Sum of rak ad ullity Let : V W be a liear trasformatio from a -dimesioal ector space V i.e. the dimdomai of is ito a ector space W. he rak ullity i.e. dimrage of dimkerel of dimdomai of You ca image that the dimdomai of should equals the dimrage of origially But some dimesios of the domai of is absorbed by the zero ector i W So the dimrage of is smaller tha the dimdomai of by the umber of how may dimesios of the domai of are absorbed by the zero ector, which is eactly the dimkerel of 6.5

36 Pf: Let be represeted by a m matri A, ad assume rak A r rak dimrage of dimcolum space of A rak A r ullity dimkerel of dimull space of A r rak ullity r r accordig to hm. 4.7 where raka + ullitya = Here we oly cosider that is represeted by a m matri A. I the et sectio, we will proe that ay liear trasformatio from a -dimesioal space to a m-dimesioal space ca be represeted by m matri 6.6

37 E 8: Fidig the rak ad ullity of a liear trasformatio Fid the rak ad ullity of the liear trasformatio : R R defie by Sol: A rak rak A ullity dimdomai of rak he rak is determied by the umber of leadig s, ad the ullity by the umber of free ariables colums without leadig s 6.7

38 E 9: Fidig the rak ad ullity of a liear trasformatio Sol: 5 7 Let : R R be a liear trasformatio a Fid the dimesio of the kerel of of the rage of is b Fid the rak of if the ullity of is 4 c Fid the rak of if ker { } a dimdomai of 5 if the dimesio dimkerel of dimrage of 5 b rak ullity 54 c rak ullity

39 Oe-to-oe: A fuctio : V W is called oe-to-oe if the preimage of eery w i the rage cosists of a sigle ector. his is equialet to sayig that is oe-to-oe iff for all u ad i V, u implies that u oe-to-oe ot oe-to-oe 6.9

40 heorem 6.6: Oe-to-oe liear trasformatio Let : V W be a liear trasformatio. he is oe-to-oe iff ker { } Pf: Suppose is oe-to-oe he ca hae oly oe solutio : i.e. ker { } Due to the fact that = i hm. 6. Suppose ker ={ } ad u= u u is a liear trasformatio, see Property i hm. 6. u ker u u is oe-to-oe because u implies that u 6.4

41 E : Oe-to-oe ad ot oe-to-oe liear trasformatio a he liear trasformatio : M M gie by A A m m is oe-to-oe because its kerel cosists of oly the m zero matri b he zero trasformatio : R R is ot oe-to-oe because its kerel is all of R 6.4

42 Oto: A fuctio : V W is said to be oto if i W has a preimage i V eery elemet is oto W whe W is equal to the rage of heorem 6.7: Oto liear trasformatios Let : V W be a liear trasformatio, where W is fiite dimesioal. he is oto if ad oly if the rak of is equal to the dimesio of W rak dimrage of dim W he defiitio of the rak of a liear trasformatio he defiitio of oto liear trasformatios 6.4

43 heorem 6.8: Oe-to-oe ad oto liear trasformatios Let : V W be a liear trasformatio with ector space V ad W both of dimesio. he Pf: is oe-to-oe if ad oly if it is oto If is oe-to-oe, the ker { } ad dimker If dimrage Cosequetly, dimker hm.6.5 hm.6.5 dimker is oto dim W is oto, the dimrage of dim W dimrage of Accordig to the defiitio of dimesio o p.7 that if a ector space V cosists of the zero ector aloe, the dimesio of V is defied as zero ker { } herefore, is oe-to-oe 6.4

44 E : m he liear trasformatio : R R is gie by A. Fid the ullity ad rak of ad determie whether is oe-to-oe, oto, or either a A Sol: b A = dimr = = dimrage of = # of leadig s c A = = dimker d A If ullity = dimker = If rak = dimr m = m :R R m dimdomai of rak ullity - oto a :R R Yes Yes b :R R Yes No c :R R No Yes d :R R No No 6.44

45 Isomorphism : A liear trasformatio : V W that isoe to oe ad oto is called a isomorphism. Moreoer,if V ad W are ector spaces such that there eists a isomorphism fromv tow, the V ad W are said to be isomorphicto each other heorem 6.9: Isomorphic spaces ad dimesio wo fiite-dimesioal ector space V ad W are isomorphic if ad oly if they are of the same dimesio Pf: Assume that V is isomorphictow, where V has dimesio here eists a L.. : V is oe-to-oe dimker W that is oe to oe ad oto dimv = dimrage of dimdomai of dimker 6.45

46 is oto dimrage of dim W hus dim V dim W Assume that V ad W both hae dimesio Let B,,, be a basis of V ad B' w, w,, w be a basis of W c c c he a arbitrary ector i V ca be represeted as ad you ca defie a L.. : V W as follows w c w c w c w by defiig w i i 6.46

47 Sice B' is a basis for V, {w, w, w } is liearly idepedet, ad the oly solutio for w= is c =c = =c = So with w=, the correspodig is, i.e., ker = {} is oe- to-oe By heorem 6.5, we ca derie that dimrage of = dimdomai of dimker = = = dimw is oto Sice this liear trasformatio is both oe-to-oe ad oto, the V ad W are isomorphic 6.47

48 Note heorem 6.9 tells us that eery ector space with dimesio is isomorphic to R E : Isomorphic ector spaces he followig ector spaces are isomorphic to each other a 4 R 4-space b c M space of all 4 matrices 4 M space of all matrices d P space of all polyomials of degree or less 5 e V {,,, 4,, i are real umbers}a subspace of R 6.48

49 Keywords i Sectio 6.: kerel of a liear trasformatio rage of a liear trasformatio rak of a liear trasformatio ullity of a liear trasformatio oe-to-oe oto Isomorphism oe-to-oe ad oto isomorphic space 6.49

50 6. Matrices for Liear rasformatios wo represetatios of the liear trasformatio :R R :,,,, 4 A 4 hree reasos for matri represetatio of a liear trasformatio: It is simpler to write It is simpler to read It is more easily adapted for computer use 6.5

51 heorem 6.: Stadard matri for a liear trasformatio m Let : R R be a liear trtasformatio such that a a a a a a e, e,, e, a a a m m m e where { e, e,, } is a stadard basis for R. he the m matri whose colums correspod to e, a a a a a a A am am am e e e, is such that = A for eery i R, A is called the Stadard matri for i 6.5

52 Pf: e e e is a liear trasformatio e e e A If A e e e, the a a am a a a m a a a m e e e e e a a am a a a m a a a e m 6.5

53 a a a a a a a a a m m m e e e herefore, A for each i R Note heorem 6. tells us that oce we kow the image of eery ector i the stadard basis that is e i, you ca use the properties of liear trasformatios to determie for ay i V 6.5

54 E : Fidig the stadard matri of a liear trasformatio Fid the stadard matri for the L.. : R R defied by, y, z y, y Sol: Vector Notatio Matri Notatio e,,, e e,,, e e,,, e 6.54

55 6.55 A e e e Note: a more direct way to costruct the stadard matri z y z y A y y z y z y A i.e.,,,, y z y y Check: he first secod row actually represets the liear trasformatio fuctio to geerate the first secod compoet of the target ector

56 E : Fidig the stadard matri of a liear trasformatio he liear trasformatio : R R is gie by projectig each poit i Sol: Notes: R, y, oto the - ais. Fid the stadard matri for e e,, A he stadard matri for the zero trasformatio from R ito R m is the m zero matri he stadard matri for the idetity trasformatio from R ito R is the idetity matri I 6.56

57 Compositio of :R R m with :R m R p :, R his compositio is deoted by heorem 6.: Compositio of liear trasformatios Let : R R ad : R R be liear trasformatios m m p with stadard matrices A ad A,the p he compositio : R R, defied by, is still a liear trasformatio he stadard matri A for is gie by the matri product A A A 6.57

58 6.58 Pf: is a liear trasformatio Let ad be ectors i ad let be ay scalar. he R c u is the stadard matri for A A u u u u u c c c c c A A A A A Note:

59 E : he stadard matri of a compositio Let ad be liear trasformatios from R ito R s.t., y, z y,, z, y, z y, z, y Fid the stadard matrices for the compositios ad ' Sol: A stadard matri for A stadard matri for 6.59

60 6.6 stadard matri for he ' matri for stadard he A A A ' A A A

61 Ierse liear trasformatio: If : R R ad : R R are L.. s.t. for eery i R ad he is called the ierse of ad issaid to be iertible Note: If the trasformatio is iertible, the the ierse is uique ad deoted by 6.6

62 heorem 6.: Eistece of a ierse trasformatio Let : R R be a liear trasformatio with stadard matri A, he the followig coditio are equialet Note: is iertible is a isomorphism A is iertible If is iertible with stadard matri A, the the stadard matri for is A For, you ca imagie that sice is oe-to-oe ad oto, for eery w i the codomai of, there is oly oe preimage, which implies that - w = ca be a L.. ad well-defied, so we ca ifer that is iertible O the cotrary, if there are may preimages for each w, it is impossible to fid a L. to represet - because for a L., there is always oe iput ad oe output, so caot be iertible 6.6

63 6.6 E 4: Fidig the ierse of a liear trasformatio he liear trasformatio : is defied by R R 4,,,, Sol: 4 matri for stadard he A 4 4 I A Show that is iertible, ad fid its ierse

64 6.64 G.-J. E. 6 I A is is iertible ad the stadard matri for herefore A 6 A 6 6 A 6,,,, I other words, Check -,, 4 = - 7, 9, =,, 4

65 he matri of relatie to the bases B ad B : : V W a liear trasformatio B{,,, } a ostadard basis for V he coordiate matri of ay relatie to B is deoted by [ ] if ca be represeted as c c c, the [ ] A matri A ca represet if the result of A multiplied by a coordiate matri of relatie to B is a coordiate matri of relatie to B, where B is a basis for W. hat is, B B ' A [ ], where A is called the matri of relatie to the bases B ad B B c c c B 6.65

66 rasformatio matri for ostadard bases the geeralizatio of heorem 6., i which stadard bases are cosidered : Let V ad W be fiite - dimesioal ector spaces with bases B ad B ', respectiely, where B {,,, } If : V W is a liear trasformatio s.t. a a a a a a a a a,,, B' B' B' m m m the the m matri whose colums correspodto i B ' 6.66

67 A B ' is such that A [ ] for eery i V B a a a a a a [ ] B [ ] B [ ] B a a a m m m he aboe result state that the coordiate of relatie to the basis B equals the multiplicatio of A defied aboe ad the coordiate of relatie to the basis B. Comparig to the result i hm. 6. = A, it ca ifer that the liear trasformatio ad the basis chage ca be achieed i oe step through multiplyig the matri A defied aboe see the figure o 6.74 for illustratio 6.67

68 E 5: Fidig a matri relatie to ostadard bases Let : R R be a liear trasformatio defied by,, Fid the matri of relatie to the basis B{,,, } ad B ' {,,, } Sol:,,,,,,,,,,, B' B' the matri for relatie to B ad B' A, B ', B ' 6.68

69 E 6: For the L.. : R R gie i Eample 5, use the matri A to fid, where, Sol:,,, B {,,,} B B' B,,, A B'{,,,} Check:,,, 6.69

70 Notes: I the special case where V = W i.e., :V V ad B = B he matri A is called the matri of relatie to the basis B If : V V is the idetity trasformatio B {,,, }: a basis for V the matri of relatie to the basis A I B B B B 6.7

71 Keywords i Sectio 6.: stadard matri for compositio of liear trasformatios ierse liear trasformatio matri of relatie to the bases B ad B' matri of relatie to the basis B: 6.7

72 6.4 rasitio Matrices ad Similarity : V V a liear trasformatio B{,,, } a basis of V B' { w, w,, w } a basis of V A matri of relatie to B B B B w w w A ' matri of relatie to ' B' B' B B' A, ad A B B B' B' w w w P trasitio matri from B' to B B B B P B B B B B trasitio matri from to ' ' ' ' from the defiitio of the trasitio matri o p.54 i the tet book or o Slide 4.8 ad 4.9 i the lecture otes P, ad P B B' B' B 6.7

73 wo ways to get from to : direct: A'[ ] [ ] B' B' B ' idirect : P AP[ ] [ ] A' P AP B' B' B' idirect direct 6.7

74 E : Fidig a matri for a liear trasformatio Sol: Fid the matri A' for :,,,,, B' R R,, reletie to the basis B' {,,,},,,,, B' A' A',, B',, B' B' B' 6.74

75 stadard matri for matri of relatie to B {,,, } A trasitio matri from B' to B P,, P B B trasitio matri from B to B' A',,,, B' B' matri of relatie B' Sole a, + b, =, a, b =, Sole c, + d, =, c, d =, P AP Sole a, + b, =, a, b =, Sole c, + d, =, c, d = -, 6.75

76 E : Fidig a matri for a liear trasformatio Sol: Let B {,, 4, } ad B' {,,, } be basis for 7 R, ad let A be the matri for : relatie to. 7 R R B Fid the matri of relatie to B' Because the specific fuctio is ukow, it is difficult to apply the direct method to derie A, so we resort to the idirect method where A = P - AP trasitio matri from B' to B : P trasitio matri from B matri of relatie to B': to B': P,, B, 4, 7 A' P AP 7 B' B B' 6.76

77 6.77 E : Fidig a matri for a liear trasformatio Sol: ' ' For the liear trasformatio : gie i E., fid,, ad, for the ector whose coordiate matri is B B B B R R 5 7 B' B P B B A 7 4 ' B B P 7 ' or ' ' B B A

78 Similar matri : For square matrices A ad A of order, A is said to be similar to A if there eist a iertible matri P s.t. A =P - AP heorem 6.: Properties of similar matrices Let A, B, ad C be square matrices of order. he the followig properties are true. A is similar to A If A is similar to B, the B is similar to A If A is similar to B ad B is similar to C, the A is similar to C Pf for ad the proof of is left i Eercise : A I AI the trasitio matri from A to A is the I A P BP PAP P P BP P PAP B Q AQ B by defiig Q P, thus B is similar to A 6.78

79 E 4: Similar matrices a A ad A' are similar because A' P AP, where P 7 b A ad A' are similar 7 because A' P AP, where P 6.79

80 E 5: A compariso of two matrices for a liear trasformatio Suppose A is the matri for : R R relatie to the stadard basis. Fid the matri for relatie to the basis Sol: he P B ' {,,,,,,,, } trasitio matri from to the stadard,,,,,, P B B B' B matri 6.8

81 matri of relatie to B': A' P AP 4 A is a diagoal matri, which is simple ad with some computatioal adatages You hae see that the stadard matri for a liear trasformatio :V V depeds o the basis used for V. What choice of basis will make the stadard matri for as simple as possible? his case shows that it is ot always the stadard basis. 6.8

82 6.8 Notes: Diagoal matrices hae may computatioal adatages oer odiagoal oes it will be used i the et chapter for d d D d k k k k d d D d D D, d d i d D d

83 Keywords i Sectio 6.4: matri of relatie to B matri of relatie to B' trasitio matri from B' to B trasitio matri from B to B' similar matri 6.8

84 6.5 Applicatios of Liear rasformatio he geometry of liear trasformatio i the plae p.47-p. Reflectio i -ais, y-ais, ad y= Horizotal ad ertical epasio ad cotractio Horizotal ad ertical shear Computer graphics to produce ay desired agle of iew of a - D figure 6.84