too many conditions to check!!

Transcription

1 Vector Spaces

2 Aioms of a Vector Space closre Defiitio : Let V be a o empty set of vectors with operatios : i. Vector additio :, v є V + v є V ii. Scalar mltiplicatio: li є V k є V where k is scalar. The, V is called a Vector Space if the followig 8 aioms hold for ay, v, w є V ) ) ) ) ) 6) 7) 8) ( v) w ( v w) associative zero vector ( ) additive iverse v v commtative k( ( a b) ( ab ) v) a ( b ) k kv k is scalar a b a, b scalars too may coditios to check!!

3 Sbspaces Defiitio : Let V be a vector space ad let W be a sbset of V. The, W is a sbspace of V if W itself is a vector space Theorem : Sppose W is a sbset of V. The, W is a sbspace if a) The zero vector belogs to W b) For every, v є W (a + bv) є W Remark : Two trivial sbspaces of V are { } ad V itself Eample : Cosider V eqal etries, (i) U U is R ie i.e. U {(a, b, c) :a a (ii) sbspace. Let U cosist of b (a, a,a) v (b, b, b) c} We do t eed to verify that the 8 aioms of a vector space hold! all v vectors U i R with ( a b)(,,)

4 Abstract Sbspaces Eamples Eamples : (prove by showig that the coditios of a sbspace hold) P (t): polyomials of degree less tha or eqal to. Note that it is o loger a sbspace if we limit the degree of the polyomial to ; why? R :tple -tple (vector) of real mbers (a, a,, a ) M m, : m real matrices F : all real fctios f(). What abot cotios or itegrable or ( ) g differetiable fctios?

5 More Sbspaces Eamples Eample : Let V = M,. Let matrices ti W is a sbspace W be a sbset of all pper (or lower) triaglar Let W be the sbset of all symmetric matrices (all-zero matri is symmetric, liear combiatio of symmetric matrices is symmetric) W is sbspace Eample : Sbset of all polyomials of eve-power is sbspace of the space of -th degree polyomials Eample : Sbset of all cotios real fctios ad sbset of all differetiable real fctios are sbspaces of the space of all real fctios

6 Spa Defiitio : Let V be a vector space. Vectors,,. i V spa V if every vector v i V is a liear combiatio of {,,. }, i.e. Remarks : v = a + a +. + a Fid a spa for X-Y plae. Is it iqe? ) Sppose {,,. } spa V. The, for ay vector w, (assig zero weight to w) the set {w,,,., } also spas V ) Sppose {,,., } spa V ad sppose that k is a liear combiatio of some of the other s. The,the s withot k also spa V Eample : Verify that the vectors (,) ad (-,-) do t spa R vectors (,) ad (,) do while the

7 Spa Eamples ) Cosider t a) he vector space R e, e, e spa R ( a, b, c) v ae be ce b) w, w, w v ( a, b, c) cw ( b c) w ( a b) w also spa R c) (show liear However, that t ay vector b combiatio of [,,], i R,, s.t. ) [,,], b b b [,, 9] does ot spa ca' t be writte as R

8 More Spa Eamples for Abstract Sbspaces ) Cosider P (t) : all polyomials of degree Every polyomial ca be epressed as a liear combiatio of (epad abot "") {, t, t,..., t } ) Cosider v ector space M, spaed by the,,, matrices

9 Itersectios & Uios of Sbspaces Vector Space s v s+tv tv Theorem : The itersectio of ay mber of sbspaces of a vector V is a sbspace of V. Qestio : what abot their io? ot a sbspace! Give a coter eample, (hit : cosider the ad y aes) Proof: Let U & W be sbspaces of V. Clearly U, W U Now, sppose ad v U W the, v U ad, v W. Frthermore, a bv U ad a bv W (sice U ad W are sbspaces) a bv U W U W is a sbspace of V W

10 The Nll Space Theorem : The soltio set W of a homogeos system A i kows is a sbspace of R Proof : Let W be the soltio A W A ad Av a bv W Remark :Soltio set of o called ll space of A. set. sppose, v W A(a a sbspace sice does' t belog to it. bv) aa - homogeos set A bav b is ot

11 Coectio to Liear Eqatios A m matri rows : m -dimesioal vectors colms : m-dimesioal vectors Systems of Eqatios A=b ca be thoght of geometrically i as ) The A matri trasforms vector to vector b ) The vector b is a liear combiatio of colms of A (lies i a vector space called the colm space of A). So, what is a vector space? v v v b v v v b

12 Row ad Colm Space of a Matri Let A be m matri. The rows (colms) of A may be viewed as vectors i R (R m ); hece they spa a sbspace of R (R m ) called the row space (colm space) of A of Liear combiatio rows (colms) of A R(A) spa (R, R C(A) spa(c, C,..., R,..., C ) ) m Eample: Describe the colm space of : (i) (ii) 6 R - - C( A) lie Eercise : describe the row ad ll spaces of fthese matrices ti as well

13 Coectio to Liear Systems of Eqatios Defiitio : the colm space of A is the sbspace of spaed by colms of A The systems of liear eqatios A=b has ) No soltio if ad oly if b is ot i the spa of the colm space of A ) Uiqe soltio if b is i the spa of colm space of A i a iqe way ) Ifiite soltios if b is i the spa of the colm space of A i ifiite ways

14 Liear Depedece & Idepedece Defiitio : We say that the vectors v, v,..., v i V are liearly depedet if scalars a, a,..., a, ot all zero, sch that a v a v... a v Otherwise the vectors are liearly idepedet, i.e. i Eample : Is a v i i implies a the set of liearly idepedet? i vectors{, t, t i,...,, t t t soltio : () ( t) ( t ) ( t t t ) By eqatig coefficiets for powers of (t) : ; hece set is } liearly idepedet. Relatio to ll space of a matri

15 Eamples o Liear Idepedece (,9,) (,,), (,,), Let ) w v liearly depedet Sice w v idepedet liearly (,,) (,,7), (,,), ) z y w v 7 7 z y z y z y z y 7 7 z y z y z y z z y

16 Liear depedece i R (geometric view) R a) Ay vectors i are liearly depedet if ad oly if they lie o the same lie throgh origi b) Ay vectors i R are liearly depedet if ad oly if they lie o the same plae v Theorem : Sppose or more o - zero vectors v, v,.., v m are liearly depedet. The, oe of the vectors is a liear combiatio of the precedig vectors, ie i.e. k st s.t vk cv cv... ck vk Theorem :The o - zero rows of a matri i echelo form are liearly idepede t

17 Basis ad Dimesio Defiitio : A set S,,..., for V (vector space) if of vectors is (i) S spas V (ii) S is liearly idepede t a basis Eercise : give eample of sets of vectors that satisfy oly oe of these coditios Theorem : Let V be a vector space sch that oe basis has "m" elemets ad aother basis has " " elemets, the m ad we write dim( V Notes : ) The vector space ) For a give basis, ). has The basis of a vector space is ot iqe bt the mber of vectors i the basis is iqe dimesio by defiitio every vect or ca be writte i oly oe way as a liear combiatio of the basis vectors prove () by cotradictio!

18 Basis ad Dimesio Eamples ) R : The it vecto rs & spa R basis for R basis. Fid other bases. ) M m : The m matrices ) m e, e,..., e are liearly idepede t, dim( R ). This is called the stadard (has i (i, j) positio ad zeros else where for i m & j ) form a basis of M dim( M ) ) m E ij m P ( t) :, t, t,..., t is a basis dim The dimesio of the soltio space W of the homogeeo s system AX (i.e. ll space of A) is ( - r) which is the mber of free variables, where is the mber of kows ad r is the rak of A (mber of pivots)

19 Theorem Lt Let V be a vector space where dim( V ). The, (i) Ay ( ) or more vectors i V are liearly depedet. For eample, ( ) vectors i (ii) Ay liearly idepedet set S elemets is a basis of V R v, v,..., v are liearly depedet,,..., with (iii) Ay spaig set T of V with elemets is a basis of V (iv) Ay set of ( -) or less vectors i V does ot spa V

20 Eample:Let V The, dim( W ). R a) dim(w) φ W dim( V ). Let W be a sbspace of (a poit) b) dim (W) W is a lie throgh h origi ii c) dim(w) W is a plae throgh origi d) dim (W) W V Eample: ) Dim ( M, ) ) Dim ( pper triaglar ) ) Dim ( symmetric matrices ) ) Dim ( diagoal) V.

21 For Sbspaces of A m ) Row space R(A) or C(A ) Colm space C(A) or ) Nll space N(A) R ) Left Nll space N(A T T ) R R(A T ) R ) R Facts : ) dim (R(A)) dim (C(A)) rak " r" ) dim (N(A)) - r A A T m ) dim (N(A m T m )) m- r m Fdametal Theorem of Liear Algebra, Part I ) dim (R(A)) dim (N(A)) dim ( R ) ) dim (C(A)) T dim (N(A )) dim ( R m ) m

22 Row ad Colm Sbspaces ) The colms of m matri A are liearly idepedet whe the rak r =. There are pivots ad o free variables. Oly = is i the ll space ) The vectors v, v,.., v are a basis for R eactly whe they are the colms of a ivertible matri. Hece, R has ifiitely may differet bases (all with the same dimesio ) ) The pivot colms of A are a basis for its colm space C(A). The pivot rows of A (or its echelo form) are a basis for R(A) ) of Note : A Ek... EE A U Therefore, ll spaces of A ad U are same The maimm mber of A is eqal to the maimm mber of colms of liearly idepedet rows A. Hece, dim( R(A) ) dim( C(A) ) liearly idepedet rak ( A)

23 Comptig Bases for the Row ad Colm Spaces Procedre (se to fid a basis for the row space of A) i. Form the matri M whose rows are the give vectors ii. Row redce M to echelo form iii. The basis is give by o-zero rows of echelo matri (or the correspodig rows i the origial matri) Procedre (se to fid a basis for the colm space) i. Form matri M whose colms are the give vectors ii. Row redce M to echelo form iii. For each colm C k i the echelo matri withot pivot, delete vector k from the give list S iv. Basis = remaiig vectors i S

24 Eample 7) 9 6 ( 8) 7 8 ( (,,,,) ),,,, (, by : spaed of sbspace a be Let R W of set from the for basis a Fid,9) (,,,,6,9,7) (,, (,8, 7,,8) S W t) idepede are which s (i other word,...,, set of from the for basis a Fid 's S W ~ : Sol M ) dim( for basis & delete W,, W basis for colm space of M

25 Eample Eample ~ A B B A for basis Fid a a) R(A) Gassia elimiatio applies elemetary operatios (liear combiatios) o rows of A; Hece row sbspace stays the same for basis form a idepedet are liearly No zero rows of have same row space. ad R(A) B A B Hece, row sbspace stays the same,,

26 Eample (Cot d) b) Fid a basis for C(A) Cl Colms with pivots C,C,C are liearly l id idepedetd form basis for C(A), 7 6, 6 8 Colms of A that cotai pivots i echelo form of A c) Fid rak( A) (i) pivots rak( A) (ii) dim ( R(A) ) rak (iii) dim( Col(A) ) rak

27 Eample ) Fid the rak ad a basis for the row space ad colm space 6 6 Soltio : Redce A to echelo form A ~ ~ 6 - The o - zero rows of the echelo form of A (or first rows of A itself!) form a basis for its R(A) ad rak( A) - Pivots are i colms &. Hece, a basis for C(A) is give by first colms of A (ot its echelo form!) of

28 matri for this problem previos Repeat Eample 7 A ~ 6 ~ U ; for basis are of colms first ; for basis Aare or of rows first, C(A) A R(A) U r,, are variables free (A)) dim( r N Fid a basis for N(A)

29 Complete Soltio to A=b Fid CompleteSoltio to Eample: 8 U form: Pt i echelo Step: Determiepivot variables: Step : Determiefree variables: Determiepivot variables: Step :,,

30 free variables Write pivot variables i terms of Step : ;. free variables Write pivot variables i terms of Step : complete. : Step ll p p ll Two basis vectors for ll space of A Two basis vectors for ll space of A

31 Complete Soltio to A = b A complete complete particlar A p p A ll A p b Procedre for comptig the complete soltio to A=b ) Pt agmetedmatri A b ) ) of Determiepivot ad free variables Epress the pivot variables as the free variables(this is (this is p ) i echelo form liear combiatios ) pls a costat vector

32 The Fdametal Sbspaces Fdametal Theorem of Liear Algebra (Part II) Cosider a m matri A with rak r. The llspace is the orthogoal complemet (will be defied i Ch. ) of the row space (i R ). The left ll space is the orthogoal complemet of the colm space (i R m ). row space R dim = r AX p = b X = X p + X dim = r R m colm space ll space dim = - r AX = left ll space dim = m - r

33 Fidig Basis for N(A) f f di i d b i Fi d E l N(A) (i) for : of dimesio ad basis Fid : Eample N(A) ~ 8 6 (i) A )) ( dim(,, variables free, A N r r 7 7 ; ll

34 ii) Write dow the complete soltio for A = b for b = 7 ; 7 complete

35 of ) N(A space ll left the of dimesio ad basis the Fid T 7 6 A ~ 7 6 A T. ;., variables free )) ( dim( A N r T. ll left.

36 Liear Trasformatios Defiitio : A trasformatio T is liear iff : i) T(+v)=T()+T(v) T()+T(v) ii) T(cv)=cT(v) where c is a scalar ad,v are vectors Defiitio : Let A be a m matri. The fctio T()=y=A is called a matri trasformatio from m to R R Fact : Every matri trasformatio o R is liear (prove it!)

37 Liear Trasformatio How to determie its matri represetatio? Tv ; Tv ;...;; Tv Tv v v v U TV T UV For V to be ivertible, its colms mst be liearly idepedet (i.e. form a basis) Idea : if yo kow reslt of applyig T(.) to basis vectors, yo kow reslt of applyig T(.) to ay vector! Bt which basis to choose? To avoid matri iversio, i V = idetity matri!

38 Represetatio Theorem Let T be a liear trasformatio from R m to R. The m matri A T ( e ) T ( e) T ( e ) e i (where is the ith it vector) has the property that T()=A ad A is iqe. Applyig sccessive trasformatios T ), T ( ),... ( correspods to sccessive matri mltiplicatios... A A

39 Eamples Rotatio by agle coter clockwise cos( ) A si( ) si( ) cos( ) Reflectio abot origi (mirror image) A Projectio o -ais (drop a perpediclar) Not ivertible! Will be stdied i detail i Chapter A