Elementary Linear Algebra

Transcription

1 Elemetary Liear Algebra Ato & Rorres th Editio Lectre Set Chapter : Eclidea Vector Spaces

2 Chapter Cotet Vectors i -Space -Space ad -Space Norm Distace i R ad Dot Prodct Orthogoality Geometry of Liear Systems Cross Prodct

3 Vectors i -Space -Space ad -Space Geometric Vectors Vectors i two dimesios called -space Vectors i three dimesios called -space The directio of the arrowhead specifies the directio of the ector The legth of the arrow specifies the magitde The tail of the arrow is called the iitial poit of the ector ad the tip the termial poit We deote ectors i boldface type; a b ad x Scalars i lowercase italic type; a k ad x We idicate a ector has iitial poit A ad termial poit B as below: AB Iitial poit Termial poit

4 Defiitios If ad w are ay two ectors the the sm + w is the ector determied as follows: Positio the ector w so that its iitial poit coicides with the termial poit of. The ector + w is represeted by the arrow from the iitial poit of to the termial poit of w. If ad w are ay two ectors the the differece of w from is defied by w + (-w). If is a ozero ector ad k is ozero real mber (scalar) the the prodct k is defied to be the ector whose legth is k times the legth of ad whose directio is the same as that of if k > ad opposite to that of if k <. We defie k if k or. A ector of the form k is called a scalar mltiple. 4

5 Defiitios Vectors with the same legth ad directio are said to be eqialet. Zero ector deotes by Additio of ectors by the parallelogram or triagle rles Vector additio iewed as Traslatio 5

6 Defiitios Vector Sbtractio w w + (-). Scalar Mltiplicatio (-) - 6

7 Vector i Coordiate Systems If a ector i -space or -space is positioed with its iitial poit at the origial of a rectaglar coordiate system. The ector is completely determied by the coordiates of its termial poit; called the compoets of relatie to the coordiate system. ( ) ( ) ( ) 7

8 Vectors whose Iitial Poit is ot at the origi Sometimes a ector is positioed so that its iitial poit is ot at the origi. If the ectorpp ad termial poit PP ( x y z has iitial poit P P ( x ) ( x y y z z ) the ) ( x ( x y x y z ) y z z ) I - space the ector with iitial poit P ( x y ) ad termial poit P ( x y ) is PP ( x x y y ) EX: Fidig the Compoets of a Vector The compoets of the ector PP ad termial poit P (75 8) are has iitial poit P ( 4) PP 8

9 -Space DEFINITION If is positie iteger the a ordered tple isa seqece of real mber (... ). The set of all orderd tple is called space ad isdeote by R DEFINITION Vector ( w We idicate this by writig w.... w.... ) ad w ( w w w... w ) i R are said to be eqialet if DEFINITION if ( + w ( k (k (-... k + w... k... w w + ( ) ( ) ad w ( w w ) ) w + w... + w... w ) w... ) i R w ad if ) k isay scalar the we defie 9

10 Properties of Vectors Associatiity of ector additio

11 Liear Combiatios Defiitio 4 If w is a ector i R... w k r i R + k where k k... k the if it ca be expressed i the form k w r r are scalars. w is said to be a liear combiatio of r the ector

12 . Norm of a Vector ( ) The legth of a ector is ofte called the orm of ad is deoted by. It follows from the Theorem of Pythagoras that the orm of a ector ( ) i -space is + + P( )

13 Norm of a ector Theorem.. If is a ector i R ( a) ( b) if ad oly if ( c) k k ad if k isay scalar the : A ector of orm is called a it ector

14 Normalizig a Vector Fid the it ector that has the same directio as ( -) 4

15 The Stadard Uit Vectors Stadard it ectors i R e (...) e eery ector ( (... ) e (...)... e... + e to be ) i R e ca be expressed as (...) 5

16 Distace i R If are two poits i -space the the distace d betwee them is the orm of ector; Similarly i -space: ) ( ad ) ( z y x P z y x P ) ( z z y y x x P P P P ) (... ) ( ) ( ) ( ad defie it to be ) ( the we deote the distace betwee ad by poits i )are... ( ad )... ( if d d R

17 Calclatig Distace i R If ( - 7) ad ( 7 ) the the distace betwee ad is 7

18 . The Dot Prodct. >. < 8

19 9 Example If the agle betwee the ectors ( ) ad ( ) is 45 the 4 4 cos θ ) ( ) ( cos θ

20 The Dot Prodct Example Calclatig Dot Prodct Usig Compoets (- 5 7) (- -4 )

21 Properties of the Dot Prodct cosθ if ad if or

22 Applicatio of Dot Prodct ISBN Nmbers : a ( ) b ( ) If [(a. b) % 9 ] the ISBN is correct otherwise o.

23 . Orthogoality Defiitio Perpediclar ectors are also called orthogoal ectors. Two ozero ectors are orthogoal if ad oly if their dot prodct is zero. To idicate that ad are orthogoal ectors we write Orthogoal ectors. (- 4) ( -) Set S {i j k} is a orthogoal set i R // Elemetary Liear Algebra

24 Plaes i -Space Oe ca specify a plae i -space by giig its icliatio ad specifyig oe of its poits. A coeiet method for a plae is to specify a ozero ector called a ormal that is perpediclar to the plae. The poit-ormal form of the eqatio of a plae: (a b c) P P ( x x y y z ) P P z a(x-x ) + b(y-y ) + c(z-z ) ax+by+cz+d ; d -ax - by - cz 4

25 Example 5

26 Lie i -Space Show that i -space the ozero ector (a b) is perpediclar to the lie ax + by + c. P P P P ( x x y y) a(x-x ) + b(y-y ) ax+by+c ; c -ax - by y (ab) P(xy) P (x y ) ax+by+c x 6

27 A Orthogoal Projectio To "decompose" a ector ito a sm of two terms oe parallel to a specified ozero ector a ad the other perpediclar to a. We hae w w ad w + w w + ( w ) The ector w is called the orthogoal projectio of o a or sometimes the ector compoet of alog a ad deoted by proj a The ector w is called the ector compoet of orthogoal to a ad deoted by w proj a // Elemetary Liear Algebra 7

28 Theorem w w proj a proj a If ad a are ectors i -space or -space ad if a the proj a proj proj a a (ector compoet of alog a) a a a a a (ector compoet of orthogoal to a) a a cosθ a // Elemetary Liear Algebra 8

29 Example Let ( ) ad a (4 ). Fid the ector compoet of ad the ector compoet of orthogoal to a. a proja a a proj alog a a a a a Soltio: a Ths a + ( ) the ector compoet of proj ad the ector compoet of 4 their dot prodct is zero. a ()(4) + ( )( ) + ()() 5 proj + 5 Verify that the ector proj a a a a ( ) ( alog a is (4 ) ( orthogoal to a is a 7 ad 5 7 a ) ( ) 7 7 ) are perpediclar by showig that // Elemetary Liear Algebra 9

30 Distace Betwee a Poit ad a Lie Fid a formla for the distace D betwee poit P ( x y ) ad the lie ax + by + c. Let Q( x y ) be ay poit o the liead positio the ector ( a b) so that its iitial poit is at Q. By irteof Example5 the ector is perpediclar to the lie (Fig..8). As idicated i the figre the distace D is eqal to the legth of the orthogoal projectio of D QP proj o ; ths QP QP Bt QP ( x x y y ) QP a( x x ) + b( y y ) a + b // Elemetary Liear Algebra

31 Example Soltio: (cot.) so that a( x x ) + b( y y) D () a + b Sice the poit Q( x y) lies o the lie its coordiates satisfy the eqatio of the lie so ax + by + c or c ax Sbstittig thisexpressio i () yields the formla ax + by + c D () a + b by // Elemetary Liear Algebra

32 Example Fid the distace betwee poit ad lie Poit is (-) Lie is 4x+y+4 Distace

33 Theorem..4 (Distace Betwee a Poit ad a Plae) The distace D betwee a poit P (x y z ) ad the plae ax + by + cz + d is D ax + by a + b + cz + c + d

34 Example (Distace Betwee a Pot ad a Plae) 4

35 Example (Distace Betwee Parallel Plaes) D ax + by a + b + cz + c + d 5

36 .4 The Geometry of Liear Systems : Vector ad Parametric Eqatios of Lies i R R Sppose that l is the lie i -space throgh the poit P (x y z ) ad parallel to the ozero ector (a b c). l cosists precisely of those poits P (x y z ) for which the ector P P is parallel to that is for which there is a scalar t sch that P P t Parametric eqatios for l: x x x x + ta ta y y y y tb + tb z z z z tc + tc If lie pass throgh the origi P is origial poit 6

37 Example x x + ta y y + tb z z + tc Parametric eqatios of a lie 7

38 Example (Itersectio of a Lie ad the xy-plae) x x + ta y y + tb z z + tc 8

39 9.5 Cross Prodct Defiitio If ( ) ad ( ) are ectors i -space the the cross prodct is the ector defied by or i determiat otatio Example Fid where ( -) ad ( ). ( -7-6) ) (

40 Stadard Uit Vectors The ectors i () j () k () hae legth ad lie alog the coordiate axes. They are called the stadard it ectors i -space. Eery ector ( ) i -space is expressible i terms of i j k sice we ca write ( ) () + () + () i + j + k For example ( - 4) i j +4k Note that i i j j k k i j k j k i k i j j i -k k j -i i k -j 4

41 4 Cross Prodct A cross prodct ca be represeted symbolically i the form of determiat: Geometric iterpretatio of cross prodct: From Lagrage s idetity we hae k j i k j i + siθ

42 Theorems Theorem.4. (Relatioships Iolig Cross Prodct ad Dot Prodct) If ad w are ectors i -space the ( ) ( ) ( ) (Lagrage s idetity) ( w) ( w) ( ) w (relatioship betwee cross & dot prodct) ( ) w ( w) ( w) (relatioship betwee cross & dot prodct) Theorem.4. (Properties of Cross Prodct) If ad w are ay ectors i -space ad k is ay scalar the - ( ) ( + w) + w ( + ) w + w k( ) (k) (k) 4

43 Area of a Parallelogram Theorem.4. (Area of a Parallelogram) If ad are ectors i -space the is eqal to the area of the parallelogram determied by ad. Example Fid the area of the triagle determied by the poit () (-) ad (4). (5/) 4

44 Triple Prodct Defiitio If ad w are ectors i -space the ( w) is called the scalar triple prodct of ad w. Remarks: ( w) The symbol ( ) w make o sese. ( w) w ( ) (w ) w w w 44

45 Theorem.5.4 The absolte ale of the determiat det is eqal to the area of the parallelogram i -space determied by the ectors ( ) ad ( ) The absolte ale of the determiat det w is eqal to the olme of the parallelepiped i -space determied by the ectors ( ) ( ) ad w (w w w ) w w 45

46 Remark 46

47 How to Fid the Volme of a Parallelepiped? Example: Fid the olme of the parallelepiped of (-4) (45) ad w (4). Soltio: ( ( r w) r w w w (8) (5) + 4() 6 Ths the olme of the parallelepiped is r r V ( w) r )