Chapter Geeral Vector Spaces Real Vector Spaces Example () Let u ad v be vectors i R ad k a scalar ( a real umber), the we ca defie additio: u+v, scalar multiplicatio: ku, kv () Let P deote the set of all polyomials of the form p=ax +bx+c For example, p=, p=x, p=x+5 are i P Give ay two polyomials p= ax +bx+c ad q= a'x +b'x+c', ad a scalar k we ca also defie additio: p+q= (a+a')x +(b+b')x+(c+c'), scalar multiplicatio: kp= kax +kbx+kc () Let M be the set of all matrices The we ca also defie the additio ad scalar multiplicatio i the ordiary way
Defiitio Let V be a arbitrary oempty set of objects o which two operatios are defied, additio ad multiplicatio by scalars (umbers) By additio, we mea a rule for associatig with each pair of objects u ad v i V, a object called the sum of u ad v; by scalar multiplicatio, we mea a rule for associatig with each scalar k ad each object u i V a object ku, called the scalar multiple of u by k If the followig axioms are satisfied by all objects u, v, w i V ad all scalars k, l, the we call V a vector space ad we call the objects i V vectors: (A) If u ad v are objects i V, the u + v V (A) u + v = v + u (A) u + ( v + w) = ( u + v) + w (A4) There is a object i V, called the zero vector for V such that + u = u + = u for all u i V (A5) For each u i V, there is a object u i V, called the egative of u, such that (A6) If k R, u V, the ku V (A7) k( u + v) = ku + kv (A8) u ( k + l) u = ku + lu k( lu) = ( kl) u = u u+(-u)=(-u)+u=
Propositio Let V be a vector space ad let u V, k R The (a) u = (b) k = (c) ( ) u = u (d) If ku =, the k = or u = 4 Example The set V vector space = R with the stadard operatios of additio ad scalar multiplicatio is a The set V of all matrices with real etries with the stadard operatios of matrix additio ad matrix scalar multiplicatio It is deoted by M The set V of all m matrices with real etries uder the stadard operatios of matrix additio ad matrix scalar multiplicatio Deote by M m 4 Let V be the set of real-valued fuctios defied o the etire real lie (, ) Let f = f ( x), g = g( x) be two such fuctios i V ad let k R The ( f + g)( x) = f ( x) + g( x ), ( kf )( x) = kf ( x) The V is a vector space uder the defied operatios of additio ad scalar multiplicatio Deote by F(, ) 5 The set P of all polyomials of degree is a vector space uder the ordiary additio ad scalar multiplicatio of polyomials 6 The set V={ (x,y,z): x+4y-z=}, uder the stadard operatios of additio ad scalar multiplicatio for vectors i R is a vector space 7 Let V = { } ad defie additio ad scalar multiplicatio as follows: + = ; k = This called the trivial space
8 The set V =R where additio ad scalar multiplicatio are defied as follows: If u = ( u, u ), v = ( v, v ), the defie u + v = ( u + v, u + v ) ; k u=( ku, ) Is this a vector space? Exercise Let V=R* be the set of all positive real umbers For x,y i V ad ay scalar k, defie x+y=xy, kx=x k Determie if V is a vector space Subspaces From a give vector space we ca costruct more ew vector spaces Oe importat type of spaces obtaied from V are the subspaces of V Defiitio A oempty subset W of a vector space V is called a subspace of V if W is itself a vector space uder the additio ad scalar multiplicatio defied o V Remarks It ca be show easily that a oempty subset W of a vector space V is a subspace if ad oly if it is closed uder additio ad scalar multiplicatio, that is u, v W implies u+v W ad ku W, kv W for ay scalar k
u v u+v ku kv W Example () Let D be the set of all the diagoal matrices The D is a subspace of M () W={(x,y): x+y=} is a subspace of R Is U={(x,y): x+y-=} a subspace of R? () Let B={ (,,,), (,,,-)} Cosider W={ k(,,,)+l(,,,-): k,l R} The W is a subspace, which is called the subspace spaed by B (4) P is a subspace of P Geerally, P m is a subspace of P if m< (5) Every V has at least two trivial subspaces, V itself ad the zero space {} (6) If Ax= is a homogeeous liear system of m equatios i ukows, the the set of solutio vectors is a subspace i R 4 Defiitio A vector w is called a liear combiatio of the vectors v, v, K, v r, if w = k v + k v + K + k r v r, for some scalars k, k, K, k r R 5 Defiitio Let S ={ v, v, K, vr} be a set of vectors i a vector space V Let W be the subspace of V cosistig of all liear combiatios of the vectors i S The W is called the space spaed by v, v, K, v r, ad we say that the vectors v, v, K, v r spa W We write W =spa ( S ) or W =spa { v, v,, } K v r 6 Remarks ()W =spa { v, v, K, vr} is the smallest subspace of V that cotais v, v, K, v r i the sese that every subspace of V that cotais v, v, K, v r must cotai W
() If S ={ v, v, K, vr} ad S ={ w w w,, K, k} are two sets of vectors i a vector space V, the Spa(S)=Spa(S') if ad oly if each vector i S is a liear combiatio of those i S, ad coversely, each vector i S is a liear combiatio of those i S Exercise Show that each of the followig sets are subspaces of the vector space M (a) The set of symmetric matrices (b) The set of lower triagular matrices (c) The set of diagoal matrices Show that the followig are subspaces of F(, : ) (a) The set of cotiuous fuctios (b) The set of differetiable fuctios Cosider the vectors u= (, ) ad v =( 64,,) i R Show that w =(,, 9 7 ) is a liear combiatio of u ad v, ad that w =( 4, 8,) is ot a liear combiatio of u ad v 4 Determie whether v = (,,), v = (,,), v = (,,) spa the vector space R Liear Idepedece For a give space V, geerally there are more tha oe subset that spas V However, amog all such spaig subsets there exist miimal oe that cotais the smallest umber of vectors Such a miimal spaig set is called a basis of the space V
Defiitio Let S ={ v, v, K, v } be a oempty set of vectors The the vector equatio r k v + k v + Kk r v r = has at least oe solutio, amely k =, k =, K, k r = If this is the oly solutio, the S is called a liearly idepedet set If there are other solutios, the S is called a liearly depedet set Examples () { (,), (,)} is liearly idepedet i R I geeral, {(,,,), (,,,,), (,,,)} is liearly idepedet i R () {,x,x,, x } is liearly idepedet i P () {,,, } is liearly idepedet i M Propositio A set S with two or more vectors is (a) Liearly depedet if ad oly if at least oe of the vectors i S is expressible as a liear combiatio of the other vectors i S (b) Liearly depedet if ad oly if o vector of S is expressible as a liear combiatio of the other vectors i S 4 Examples () If v = (,,,), v = (,, 5, ), v = ( 7, 58,,), show that S={ v, v, v } is liearly depedet () Determie whether v = (,,), v = (,, 56 ), v = (,, ) form a liearly depedet set i R
() Show that the fuctios f = x, f = si x form a liearly idepedet set of vectors i F(, ) Exercise Show that the polyomials p = x, p = 5+ x x, p = + x x form a liearly depedet set i P Prove: (a) A fiite set of vectors that cotais the zero vector is liearly depedet (b) A set with exactly two vectors is liearly idepedet if ad oly if either vector is a scalar multiple of the other Let S ={ v, v, K, vr} be a set of vectors i R If r >, the S is liearly depedet 4 Prove if B is a subset of a liearly idepedet set S, the B is also liearly idepedet Is a subset of liearly depedet set always liearly depedet? 5 Let S ad B be two liearly idepedet sets of vectors i V Show that Spa{S} Spa {B} = iff B S is liearly idepedet 4 Basis ad Dimesio 4 Defiitios Let V be a vector space ad let S ={ v, v, K, vr} be a set of vectors i V The S is called a basis for V if the followig two coditios hold: S is liearly idepedet; S spas V
4 Theorem Let S ={ v, v, K, v} be a basis for a vector space V The every vector v ca be expressed as v = c v + c v + K + c v i a uique way 4 Defiitio The scalars c, c, K, c i Theorem 4 are called the coordiates of v relative to the basis S The vector ( c, c, K, c ), deoted by [v] S, i R is called the coordiate vector of v relative to S 44 Remarks For a give basis, the fuctio [] S : V R is a bijectio 45 Examples () Let e = (,,, K, ), e = (,,,, ),, e = K K (,,, K,) The S ={ e, e, K, e} is called the stadard basis for R () {,x,x,, x } is called the stadard basis of P () There is o a fiite basis for F(, ) 46 Theorem All bases for a vector spaces have the same umber of vectors 47 Defiitio () A ozero vector space is called fiite-dimesioal if it cotais a set of vectors { v, v,, v } that forms a basis If o such set exists, V is called ifiite dimesioal I K additio, the zero vector space is cosidered to be fiite-dimesioal,
() The dimesio of a fiite-dimesioal vector space V, deoted by dim(v) is defied to be the umber of vectors i a basis for V I additio, we defie the zero vector space to have dimesio zero 47 Theorem* If V is a -dimesioal vector space, the: (a) Every set with more tha vectors is liearly depedet (b) If S has vectors ad spas V, the S is a basis (c) If S is liearly idepedet ad has vectors, the S is a basis 48 Theorem * Let S be a fiite set of vectors i a fiite-dimesioal vector space V (a) If S spas V, but is ot a basis for V, the S ca be reduced to a basis for V by removig appropriate vectors from S (b) If S is a liearly idepedet set that is ot already a basis for V, the S ca be elarged to a basis for V by isertig appropriate vectors ito S 49 Examples () dim(r )= S ={ e, e, K, e} is a basis for R () Let v = (,, ), v = (,9, ), v = (,, 4) Show that the set S={ v, v, v } is a basis for R () Let S={ v, v, v } be the basis for R i () (a) Fid the co-ordiate vector of v= (, 5,9) with respect to S (b) Fid the vector v i R whose co-ordiate vector with respect to S is ( v) S = (,,) (4) S ={, xx,, K, x } is a basis for the vector space P, so dim(p )= The coordiate vector of the polyomial p= a + ax+ ax relative to the basis S ={, xx, } for P is [p] S =(a,a,a )
(5) Determie a basis for ad the dimesio of the solutio space of the homogeeous system x + x x + x = 5 x x + x x + x = 4 5 x + x x x = 5 x + x + x = 4 5 Exercise 4 Prove that S={ (,,),(,,),(,,)} is a basis of R Fid the coordiate vector of v=(,,5) with respect to S Let S ={ v, v, K, vr} be a liearly idepedet set i the vector space V The S is a basis for the subspace spa{s} Determie if S={ +x,x+x, + x } is a basis of P Fid [a x +bx+c] S if S is a basis 4 Fid the dimesio of the solutio space of the followig liear system x-y+z -w= x +z = y +w= 5 Prove if S ={ v, v,, vr} r liearly depedet 6 Prove that for ay vectors u,v ad w i a vector space V, { u-v,v-w, w-u} is liearly depedet 7 Let { v,v, v } be a basis of a space V Prove: (i) { v, v, v } is a basis (ii) { kv,v, v } is a basis for ay ozero scalar k
(iii) { v,kv +v, v } is a basis 8 Show that if { v,v,,v r } is a basis of V, the { v, v +v,, v +v + +v r } is also a basis of V 9 Show that the vector space C[,] has o fiite basis 5 Row Space, Colum Space, ad Nullspace 5 Defiitios Let A be a m matrix The the subspace of R spaed by the row vectors of A is called the row space of A, ad the subspace of R m spaed by the colum vectors of A is called the colum space of A The solutio space of the homogeeous system of equatios Ax =, which is a subspace of R, is called the ullspace of A 5 Propositio () Elemetary row operatios do ot chage the ullspace of a matrix () Elemetary row operatios do ot chage the row space of a matrix () If a matrix R is i row-echelo form, the the ozero row vectors form a basis for the row space of R 5 Propositio Let A ad B be row equivalet matrices The (a) A give set of colum vectors of A is liearly idepedet if ad oly if the correspodig colum vectors of B are liearly idepedet
(b) A give set of colum vectors of A forms a basis for the colum space of A if ad oly if the correspodig colum vectors of B form a basis for the colum space of B (c) If a matrix A is i row-echelo form, the the colum Vectors that cotais a leadig form a basis for the colum space of A 54 Remarks () By 5, to fid a basis for the row space of A, we first reduce A to its row echelo form The the rows cotaiig leadig form a basis of the row space () To fid a basis of the colum space of A, just choose those colums i A correspodig to the colums i the row echelo form of A that cotais a leadig () From () ad () it follows that the row space ad the colum space of a matrix A have the same dimesio 55 Example Let S={ (,,), (,,), (,,)} Fid a basis of the space Spa{S} Solutio: Let A= The Spa{S} is the colum space of A Reduce A to row echelo form A'=
The first ad secod colums of A' cotais leadig, so the correspodig colums i A for a basis of A, ad therefore of Spa{S}, that is { (,,),(,,)} 56 Algorithm: Give a set of vectors S ={ v, v, K, vk} i R To fid a subset of these vectors that form a basis for spa(s): Step : Form the matrix A havig v, v, K, v k as its colum vectors Step : Reduce the matrix A to its reduced row-echelo form R, ad let w, w, K, w k be the colum vectors of R Step : Idetify the colum vectors that cotais a leadig These vectors form a basis for spa(s) The correspodig colum vectors of A are the basis vectors for spa(s) 57 Defiitio Let A be a matrix The dimesio of the row space ( or the colum space) of A is called the rak of A, deoted by rak(a) The dimesio of the ullspace of A is called the ullity of A, deoted by ullity(a) 58 Theorem( Dimesio theorem for matrices) Let A have colums The rak(a) + ullity(a) = 59 Theorem* Give A with correspodig liear trasformatio T:R R, where T(u)=Au The the followig statemets are equivalet (a) A is ivertible (b) Ax = has oly the trivial solutio (c) The reduced row-echelo form of A is I (d) A is a product of elemetary matrices (e) Ax = b is cosistet for every matrix b
(f) Ax = b has exactly oe solutio (g) det(a) (h) The rage of T is R (i) T is oe-oe (j) The colum vectors of A are liearly idepedet (k) The row vectors of A are liearly idepedet (l) The colum vectors of A spa R (m) The row vectors of A spa R () The colum vectors of A form a basis for R (o) The row vectors of A form a basis for R (p) rak(a) = (q) ullity(a) = Exercise 5 Fid a basis for the space spaed by the vectors v = (,,,,), v = (, 5,, 6, ), v = ( 55,,,, ), v = ( 6886,,,, ) 4 Fid the rak ad ullity of the matrix A= Verify the result that rak(a) + ullity(a) = Let A be a 7 4 matrix Show that the row vectors of A must be liearly depedet 4Give a example to show that elemetary row operatios may chage the colum space of a matrix
5 Prove that a system of liear equatio Ax = b is cosistet if ad oly if b is i the colum space of A
Chapter Liear trasformatios Defiitio ad basic properties Defiitio A mappig T: V W from a vector space V ito a vector space W is called a liear trasformatio from V to W if for ay two vectors u, v i V ad ay scalar c, we have (a) T(u+v)=T(u)+T(v), ad (b) T(cu)=cT(u)
u T T(u) V W A liear trasformatio T:V V from V ito itself is called a deote the set of all liear operators o V liear operator o V We use L(V) to If a liear trasformatio T:V W is a bijectio( ie ijective ad surjective), it is called a liear isomorphism If there exists a isomorphism betwee V ad W we say that V ad W are isomorphic
Example () The mappig T: R R defied by T(x,y)=(x-y,x,x+y) is a liear trasformatio Note that if we write each vector as a oe colum matrix, the for each u R, T(u)= Au, where A= x ad u= y () Let C[,] be the vector space of all cotiuous real valued fuctios defied o [,] Defie the map T:C[,] R by
T(f)= f ( x ) dx The T is a liear trasformatio from C[,] to R () Let P be the vector space of real polyomial fuctios of degree less tha or equal to, of the form f(x)=a x +a x + +a x+ a For each f P, let T(f(x))=f'(x), the derivative of f(x) The T: P P is a liear trasformatio (4) Let V be a vector space with dimesio, ad let B={ b,b,,b } be a basis of V Defie a mappig T: V R T(u)=(u,u,,u ), by which is the coordiate vector of u with respect to the basis B The T is a liear trasformatio, ad it is actually a isomorphism Usually we use [] B to deote this liear isomorphism From this it follows that every -dimesioal space is isomorphic to R Hece very two -dimesioal vector spaces are isomorphic
(5) The mappig T:R R defied by T(u)=u+(,) is ot liear Theorem Let T:V W be a liear trasformatio The () T()=; () T(a u +a u + +a m u m )=a T(u )+a T(u )+ +a m T(u m ), for ay vectors u i ad scalars a i (i=,,); () T(-u)=-T(u) 4 Defiitio Let T:V W be a liear trasformatio () The kerel of T, deoted by ker(t), is the set
ker(t)={u V: T(u)= } () The rage of T, deoted by R(T), is the set R(T)={ T(u): u V} The rak of T, deoted by rak(t), is defied to be the dimesio of R(T) The ullity of T deoted by ullity(t), is defied to be the dimesio of ker(t) V Ker(T) W
R(T) V W 5 Example Let T A : R R m be the liear trasformatio defied by T A (u)=au, where
A is a m matrix The ker(t A ) is the ullspace of A ad R(T A ) is the colum space of A 6 Theorem(Dimesio Theorem for liear trasformatios}) the Let T:V W be a liear trasformatio from a -dimesioal space V to a m-dimesioal space W, rak(t)+ullity(t)= Exercise Determie which of the followig mappigs are liear trasformatios () T: P R, where T seds a polyomial p= a x +a to the vector T(p)=( a, a + a,, a +a + +a + a ) x + +a x+ a () T: C[,] R, where for each f(x) i C[,],
T(f)= f ( x) h( x) dx, where h(x) is a fixed cotiuous fuctio () T: R R, where T(u)=(,,) for each u=(x,y,z) i R Let T ad F be two liear trasformatios from V to W ad k ad l be scalars Show that kt+lf is also a liear trasformatio, where (kt+lf)(u)=kt(u)+lf(u) for every u i V Show that all the liear trasformatios from V to W form a vector space with respect to the additio ad scalar multiplicatio of liear trasformatios Let M ( R) be the vector space of all real matrices Let A be a fixed by matrix Defie σ: M ( R) M ( R) by σ (X)=AX-XA for each X i M ( R) () Show that σ is liear () Prove for ay X,Y i M ( R), σ (XY)= σ (X)Y+Xσ (Y) 4 For ay α, β i L(V), defie [ α, β ]=αβ - β α Show that for ay α, β, γ i L(V), the followig equatio holds
[[α,β],γ]+[[β,γ],α]+[[γ,α],β]= Matrices of liear trasformatios Theorem Let T: R R m be a liear trasformatio ad let B={ e,e,,e } be the stadard basis of R Let A=[ T(e ) T(e ) T(e )] be the m matrix with colum vectors T(e i )(I=,,,) The T=T A The above matrix A determied by T is called the stadard matrix of T Now let T: V W be ay liear trasformatio ad B ad S be a give basis of V ad W respectively, the there exist liear isomorphisms
[] B : V R ad [] S :W R m, here we assume V ad W have dimesio ad m respectively The the followig compositio of three liear trasformatios is liear, R [] B V T W [] S R m By theorem there is a matrix A such that T A equals this compositio This matrix A is uiquely determied by T ad is called the matrix of A with respect to the bases B ad S Defiitio Let T: V W be a liear trasformatio from the -dimesioal space V ito m-dimesioal space W, ad let B ad S be bases of V ad W respectively The uique matrix A that makes the followig diagram commutes is called the matrix of T with respect to the give bases B ad S, ad is deoted by A=[T] S, B
V T W [] B [] S R T A R m If T:V V is a liear operator ad B=B', the we write [T] B ', B =[T] B Examples Let T: P P be the liear trasformatio defied i example(), B={,+x,, +x+ +x } ad S={,x,, (-)x } The B ad S are bases of P ad P respectively Fid the matrix [T] S, B
Solutio Let H = [] S T ([] B ) A=[ H(e ) H(e ) H(e )] : R R m The by theorem, the required matrix H(e )=H((,,,))= [] S T ([] B ) ((,,,))= [] S T()= [] S ()=(,, ) H(e )= [] S T ([] B ) H(e )=[] S T ([] B ) ((,,,,))= [] S T (+x)= [] S ()=(,,,,) ((,,,,,))= [] S T (+x+x )=[] S (+x)=(,,,,) H(e )=(,,,) Thus [T] S, B =A=[ H(e ) H(e ) H(e )] =
Remark: From the above example it ca be see that, i geeral, if T=( u, u,, u ), the [T] S, T =[ [T(u ) ] S [T(u ) ] S [T(u ) ] S ] Exercise Let T: R R 4 be give by T(x,y,z,w)=( x+y,y+z,z+w,w+x) Fid [T] B,B', where B={(,,),(,,),(,,)} ad B'={(,,,), (,,,,),(,,,),(,,,)} ( x Let T: P P be the liear operator T(f)=f' Fid [T] S, where S={, x-,! ) ) ( x,,! } Suppose T: R R is a liear trasformatio such that 5 [T] S = 8 5 7 5 8, 6
where S ={ u, u, u } is a basis of R Fid [T] B, where B ={ u,+u + u, u +4 u + u, u,+u + u } 4 Let S={ γ, γ,, γ } be a basis of a vector space V Suppose α = γ, β j = bγ ij i j a ij i= i i=, j =,,, ad { α, α,, α } is liearly idepedet Let T: V V be a liear operator such that T( α)= i β, i i=,,, Fid [T] S Similar matrices Problem: Give ay liear operator T: V V o V ad two bases B ad B' of V, we have two matrices: [T] B ad [T] B ' What is the relatio betwee these two matrices?
Defiitio Let B ad B' be two bases of a fiite dimesioal space V There is a uique ivertible matrix P, called the trasitio matrix from B' to B, satisfyig the coditio for all vector u i V [u] B =P[u] B' Propositio Let B={u,u,,u } ad B'={u ',u ',,u '} be two bases of V The the trasitio matrix from B' to B is P=[ [u '] B [u '] B [u '] B ] Example Cosider the bases B={,x,x } ad B'={,+x,+x+x } of P Fid the trasitio
matrix from B' to B Solutio: By the above result P=[[] B [+x] B [+x+x ] B ] [] B =(,,), [+x] B =(,/,), [+x+x ] B ]=(,/,/) Hece P= / / / Theorem 5 Let T:V V be a liear operator o a fiite dimesioal vector space V, ad let B ad B' be bases for V
The [T] B =P [T] B' P, where P is the trasitio matrix from B' to B Defiitio 6 Two matrices A ad B are said to be similar to each other if there is a ivertible matrix P such that A=P AP Remarks 7 () If A is similar to B, the det(a)=det(b) rak(a)=rak(b)
ullity(a)=ullity(b) tr(a)=tr(b) A ad B have the same characteristic polyomial A ad B have the same eigevalues () If A ad A' are two matrices of a liear operator T:V V with respect to two bases of V, the A ad A' are similar Exercise Prove if A is similar to B the (a) det(a)=det(b); (b) rak(a)=rak(b);
(c) tr(a)=tr(b) Prove if A is similar to B ad A is ivertible, the B is ivertible ad A is similar to B Let = = = b a c a c b c b a C a c b c b a b a c B c b a b a c a c b A,, Show that A,B ad C are similar to each other 4Prove that the followig two diagoal matrices are similar if ad oly if b b b,,, is a re-arragemet of a a a,,,, b b b a a a, 4 Ivariat subspaces
Give a liear operator T o V, we wish to fid a basis B of V such that [T] B is i a simpler form, such as a diagoal matrix This problem is closely related to ivariat subspace problem Defiitio 4 Let T:V V be a liear operator A subspace W of V is said to be ivariat uder T if T(W) W The W is called W a ivariat subspace of T Example 4 {} ad V are ivariat uder ay T These are called the trivial ivariat subspaces of T Let T(u)=u for each u i V The every subspace of V is ivariat uder this liear operator If W is a oe dimesioal subspace with a basis B={ u },the W is a ivariat subspace of T if ad oly if T(u)=cu for some scalar c Remark 4 Suppose that W is a ivariat subspace of T Chose a basis B={ α,α,,α k, α k+,, α } of V so that { α,α,,α k } is a basis of W The A A [T] B = A where A i are submatrices,
I particular, if V= W W W k, the we ca chose a basis B of V such that A [T] B = A, A k where the sizes of A i s are determied by the dimesio of W i s If each of the W i s has dimesio oe, the [T] B becomes a diagoal matrix Exercise 4 Show that if A ad B are ivariat subspaces of T:V V, the A+B ad A B are also ivariat Spaces of T, where A+B={ u+v: u A, v B}
Chapter Eigevalues ad eigevectors Eigevalues ad eigevectors of matrices a Suppose A= c b d λ is similar to a diagoal matrix D= ad A=PDP Let P=[ X Y ] where X ad Y are colum vectors i R The AP=DP Thus [AX AY]=[λX Y], ad AX=λX, AY=Y The scalars λ, are called eigevalues ad X ad Y are called eigevectors of A
Defiitio Let A be a matrix A o-zero vector x R is called a eigevector of A if there is a scalar λ such that Ax= λ x The umber λ is called a eigevalue of A, ad the vector x is called a eigevector of A correspodig to λ Lemma A scalar λ is a eigevalue of A if ad oly if it is a solutio of the equatio det( λ I-A)= Defiitio The equatio det( λ I-A)= is called the characteristic equatio of A The polyomial i variable λ obtaied by expadig det( λ I-A) is called the characteristic polyomial of A Geerally, det( λ I-A)= λ +c λ where c =(-) det(a) ad c =tr(a) ++c λ +c,
Theorem 4(Cayley-Hamilto) If λ +c λ ++c λ+c = is the characteristic equatio of A, the A +c A where I is the idetity matrix ++c A+c I=, Example 4 Fid all the eigevalues of A, where A= 4 7 8 Solutio (i)the characteristic polyomial of A is det(λi-a)= λ -8λ +7λ-4 (ii) To solve λ -8λ +7λ-4= we first try if it has ay iteger solutio
Ay iteger solutio must be a divisor of the costat -4 By checkig all the divisors,-,,-,4,-4, we fid that 4 is a solutio The λ -8λ +7λ-4=(λ-4)(λ -4λ +) The remaiig eigevalues are the solutios of λ -4λ +=, they are λ =+ ad λ =- So 4, λ =+ ad λ =- are the eigevalues of A Defiitio 5 Let λ be a eigevalue of A the solutios of the liear system (λ I-A)x= is a subspace of R, it is called the eigespace of A Example 6 Fid a basis for each of the eigespaces of A=
Solutio: The eigevalues of A are λ = ad λ = So there are two eigespaces of A () The eigespace of A correspodig to λ = is the solutio space of (I-A)X=, that is z y x = The geeral solutio of this system is x =-s, y =t, z =s So S={ (-,,), (,,)} is a basis of the eigespace correspodig to λ = () The eigespace of A correspodig to λ = is the solutio space of the liear system z y x =
The geeral solutio of this system is x =-s, y =s, z =s Thus B={(-,,)} is a basis for the eigespace of A correspodig to λ = Exercise 5 Let A= 7 (i) Fid the characteristic equatio of A (ii) Fid all the eigevalues of A
(iii) Fid a basis for each of the eigespaces a b Let A= ad λ + k λ +m be its characteristic polyomial Verify that c d m=det(a) ad k=-tr(a) Suppose that λ ad are two differet eigevalues of a matrix A, ad u is a eigevector with respect to λ ad v is a eigevector with respect to Prove that { u,v} is liearly idepedet Geeralize this result Diagoalizatio problem I this sectio we ivestigate : () whe a give matrix A is diagoalizable, ad () how to fid a matrix P that diagoalizes A Defiitio A square matrix A is said to be diagoalizable if there is a ivertible matrix P such that P AP is a Diagoal matrix; the matrix P is said to diagoalize A Theorem
If A is a matrix, the A is digoalizable iff the sum of the dimesios of all eigespaces of A is Algorithm Steps for fidig P Step Fid all eigevalues of A, say λ, λ,, λ r Step Fid a basis for each of the eigespaces Step If the sum of the dimesios of all eigespaces is the A is diagoalizable, ad P is obtaied by puttig Step 4 Suppose all vectors i each of the bases as colum vectors q, q,,q are the vectors from all the bases correspodig to the eigevalues,,, The P AP=D, where D is the diagoal matrix of which the etries o the mai diagoal are,,, Example 4 The matrix A= has two eigevalues λ = ad λ =
The dimesio of the eigespace correspodig to is ad that correspodig to is Thus A is diagoalizable {(-,,),(,,)} ad {(-,,)} are the bases for the two eigespaces Thus P= diagoalizes A, ad P AP= Theorem 5 If a matrix A has differet eigevalues, the A is diagoalizable Example 6 Every triagular matrix with differet etries o its mai diagoal is diagoalizable Defiitio7 Let λ be a eigevalue of a matrix A The dimesio of the eigespace correspodig to λ is called the geometric multiplicatio ofλ, ad the umber of times that λ -λ appears as a factor i the characteristic polyomial of A is called the algebraic multiplicity of λ
Theorem 8 Let A be a square matrix The () For ay eigevalue of A the geometric multiplicity is less tha or equal to the algebraic multiplicity () A is diagoalizable if ad oly if the geometric multiplicity is equal to the algebraic multiplicity for each eigevalue of A Exercise a b x z Let A= be diagoalized by P= c d such that y w λ AP= P (i) Verify that λ ad are eigevalues of A
x z (ii) Let u= ad v= y Verify Au=λu ad Av=v w Show that if B={ u, u,, u k } cosists of eigevectors of a matrix A, the the subspace spa(b) is ivariat uder T A : R R Eigevalues ad eigevectors of liear operators Defiitio Let T: T: V V be a liear operator o a vector space V (i) A ozero vector u i V is called a eigevector of T if Tu=λu for some scalar λ The scalar λ is the called a eigevalue of T, ad u is called a eigevector of T correspodig to λ (ii) The eigespace of T correspodig to a eigevalue λ is the kerel of λi -T: V V
Propositio Let T:V V be a liear trasformatio o a fiite dimesioal space V ad B is a basis for V The () The eigevalues of T are the same as the eigevalues of the matrix [T] B () A vector u is a eigevector of T correspodig to λ if ad oly if its coordiator vector [u] B is a eigevector of [T] B correspodig to λ Example Let T:P P be the liear operator o P defied by T(a+bx+cx )=(a+b+c)+(b+c)x+cx Fid all eigevalues of T ad a basis of the eigespace for each eigevalue Solutio: We chose to use the stadard basis B={,x,x } of P The matrix of T with respect to B is
[T] B = A has oly oe eigevalue λ= A polyomial p(x)=a+bx+cx is a eigevector of T correspodig to λ= if ad oly if the vector [p(x)] B =(a,b,c) is a solutio of the liear system (I-A)X= The dimesio of the eigespace is, ad S={(,, )} is a basis for the eigespace Thus the eigevecotors of T with respect to λ= are all costat polyomials Example 4 Let D be the vector space of all ifiitely differetiable fuctios defied o R, ad δ : f(x)a f (x) is the differetiatio operator o D The for each real umber λ, we have δ (e λ x )=λe λ x Thus every real umber is a eigevalue of δ, ad the fuctio e λx is a eigevector correspodig to λ Exercise Let C[,] be the vector space of cotiuous real fuctios defied o [,] Defie F: C[,] C[,] by F(f(x)=xf(x), for all f(x) C[,]
() Prove that F is a liear operator () Prove that F has o eigevalue Prove: if a liear operator T:V V is ivertible the is ot a eigevalue of T
Chapter 4 Orthogoality i -space 4 Dot Product Defiitio 4 Give two vectors u = ( u, u,, ), v = ( v, v,, ) defied as L u L v u v = u v + u v + L+ u v, the dot product of u ad v is Defiitio 4 For ay vector u = ( u, u,, ) L u, the orm of u is defied to be u = u u = u + u + L+ u Theorem 4 For ay vectors u (a) u v = v u, v, w R ad scalar k, we have (b) ( u v) w = u w + v w + (c) ( u) v = k( u v) k Defiitio 44 () Two vectors of R are orthogoal if their dot product is zero () A set B of vectors i R is called a orthogoal set if every two differet vectors i B are orthogoal () A set B of vectors i each u B R is called orthoormal if it is orthogoal ad = u for Example 45 If u is orthogoal to every ( i =, m) v i,,, the u is orthogoal to k v + k v + L+ k m v m
Defiitio 46 Let W be a subspace of R A vector u is said to be orthogoal to W if it is orthogoal to every vector i W The set of all vectors which are orthogoal to W is called the orthogoal complemet of W, ad is deoted by W Example 47 I u is orthogoal to B = {(,, )(,,, 6 ) } = (,, ) R, = (,, ) v is ot orthogoal to B Theorem 48 Let W be a subspace of () R The W is a subspace of () W W = {} () ( W ) = W R Exercise 4 Let B be a set of vectors i u is orthogoal to B R Show that W, where =Spa (,, )(,,, ) u W, where W= Spa( B ), if ad oly if { } Fid the dimesio of W Prove for ay subspace W of R, W W = {} 4 Orthoormal Bases Let = {(,, )(,,, )(,,, ) } B The B is a basis of R which is a orthoormal set S =,,,,,,,, is also such a basis Let c be ozero umber ad u be a vector We show write c u to deote u c
Defiitio 4 A basis S of R is called a orthoormal basis if S is a orthoormal set Example 4 () B =,, () If { u, u,, } ie, is a orthoormal basis of R S = L u is a basis which is orthogoal, the the ormalizatio of S, S = u u u, u u, L, u is a orthoormal basis Theorem 4 Let { v, v,, } S = L v be a basis of R The for ay u R, ( u v ) v + ( u v ) v + + ( u v ) v u = L I other words, the coordiator vector of u with respect to S is [] u = ( u v, u v,, u ) S v L Theorem 44 Every orthogoal set of vectors i R is liearly idepedet The followig algorithm provides us with a method to obtai a orthoormal basis from ay give basis Algorithm 45 (Gram-Schmidt Process) Give a basis { u, u,, } L u of a subspace W of Step : Let v = u u v Step : Let v = u v v Step : Suppose v, R, v, L v k have bee defied, the u v u v u v + L v k + k + k + k v k = u k + v v v v v k Step 4: The ormalize { v, v,, } L v we get the required orthoormal basis: k
v v v,, L, v v v Example 46 Trasform B = {(,, )(,,, )(,,, ) } ito a orthoormal basis of R Exercise 4 Verify that vectors 4 4 v =,,, v =,,, 5 5 5 5 v = (,, ) form a orthoormal basis of R The use Theorem 4 to express the followig vectors as liear combiatios of v, v, v Check your results () (,, ) ; () (, 7, 4 ) ; () ( a b, c ), Use the Gram-Schmidt process to trasform the followig bases of R ito orthoormal bases Check your results,,,,,,,, () {( )( )( ) } () {(,, )(,, 7, )(,, 4, ) } u v () Show that u v is orthogoal to v v () If { v, v,, } B = L v k is a orthogoal set of vectors i R, the for ay u R u is orthogoal to B v u v u v v k L v k v v v k u v 4 Least Squares Solutios of a Liear System Least Squares Problem Let Ax = b be a liear system of m equatios i variables Fid a vector x such that A x b A x b holds for ay vector x i R Such a x, if exists, is called a least squares solutio of the give liear system A liear system may ot have a exact solutio, but it always has a least squares solutio
Theorem 4 For ay liear system Ax = b, the liear system T T A Ax = A b is cosistet, ad all solutios of this liear system are least squares solutios of Ax = b Theorem 4 If A is a m matrix with liearly idepedet colum vectors, the for every vector b, the liear system Ax = b has a uique least squares solutio give by T T ( A A ) A b x = Example 4 Fid the least squares solutio of the followig liear system: x y = x y = x + y = Exercise 4 Fid the least squares solutios of the followig liear systems Ax = b 7 () A =, b = 7 () A =, b = Prove if Ax = b is cosistet, the every least squares solutio of Ax = b is a exact solutio 44 Orthogoal Diagoalizatio Example 44 Cosider the rotatio of R through a agle of θ This is a liear trasformatio T : R R, such that ( x, y ) ( x cosθ y siθ, x si θ y cosθ ) T = +
The stadard matrix of T is It is easy to check that A T A = I, ie cosθ siθ A = siθ cosθ A T = A Moreover det ( A ) = Defiitio 44 A square matrix P is called a orthogoal matrix if P T P = I That is, P T is the iverse of P Example 44 A liear trasformatio ad det ([ T ]) = T : R R is a rotatio if ad oly if [ ] This is also true for rotatios o R T is a orthogoal matrix Defiitio 444 A liear trasformatio T R R orthogoal ad det([t])= : is called a rotatio if the stadard matrix [ ] T of T is Problem Give a square matrix A () Is there a orthogoal matrix P such that P T AP () How to fid such a P? is a diagoal matrix? Defiitio 445 A square matrix A is called orthogoally diagoalizable if there is a orthogoal matrix P such that is a diagoal matrix P is said to orthogoally diagoalize A P T AP = D Example 446 Matrix is orthogoally diagoalized by A =
P = Theorem 447 If A is a matrix, the the followig are equivalet: (a) A is orthogoally diagoalizable (b) A has a orthoormal set of eigevectors (c) A is symmetric Example 448 4 is orthogoally diagoalizable 4 9 4 is ot orthogoally diagoalizable 7 9 Theorem 449 If A is a symmetric matrix, the (a) The eigevalues of A are all real umbers (b) Eigevectors from differet eigespaces are orthogoal Give a symmetric matrix A, we ca use the followig algorithm to fid P that orthogoally diagoalizes A: Algorithm 44 Step : Fid a basis for each of the eigespaces Step : Apply the Gram-Schmidt process to each of these bases to obtai a orthoormal basis for each eigespace Step : Form the matrix P whose colums are the basis vectors costructed i Step ; this matrix orthogoally diagoalizes A Example 44 Fid a matrix P that orthogoally diagoalizes the followig matrix
Exercise 44 For each of the followig symmetric matrices, fid a matrix P that orthogoally diagoalizes the matrix Check your results, 6 6 Fid a matrix A ad two eigevectors from differet eigespaces of A such that they are ot orthogoal T Prove Av u = v A u for ay two vectors u ad v 4 Prove a square matrix A is orthogoal if ad oly if the row vectors (colum vectors) of A is a orthoormal set 45 Diagoalizig Quadratic Forms Defiitio 45 A quadratic form i the variables where A is a symmetric matrix x,, x, L x is a expressio that ca be writte as [ x, x, L x ], x x A M x Example 45 For ay a, b, c, ax + bxy + c = So it is a quadratic form i two variables x ad y a c b x d y [ xy]
Remark A quadratic form i variables where x,, x, L x ca be writte as X T AX, x x X = M x Defiitio 45 A quadratic form X T AX is positive defiite if u T Au for all u A symmetric matrix A is called positive defiite if the quadratic form X T AX is positive defiite Theorem 454 A symmetric matrix A is positive defiite if all eigevalues of A are positive Theorem 455 Let X T AX be a quadratic form i the variables x, x, L, x If P orthogoally diagoalizes A, ad if the ew variables y, y, L, y are defied by X = PY, the substitutig this equatio i X T AX we get where X T AX λ, λ, L, λ are the eigevalues of A T = Y DY = λ y + λ y + L + λ y, The matrix P i this theorem is said to orthogoally diagoalize the quadratic form Example 456 Fid a chage of variables that will reduce the quadratic form x x 4xx + 4xx to a sum of squares Theorem 457 (Pricipal Axes Theorem for Let ax be the equatio of a quadric Q, ad let R ) + by + cz + dxy + exz + fyz + gx + hy + iz + j = X T AX = ax + by + cz + dxy + exz + fyz
be the associated quadratic form The coordiate axes ca be rotated so that the equatio of Q i the x' y' z' -coordiate system has the form λ x' + λ y' + λz' + g' x' + h' y' + i' z' + j = where λ, λ, λ are eigevalues of A The rotatio ca be accomplished by the substitutio x = Px' where P orthogoally diagoalizes X T AX ad det ( P ) = Exercise 45 I each part, fid a chage of variables that reduces the quadratic form to a sum or differece of squares () x + x xx () x x Fid a rotatio x = Px' that removes the cross-product terms of the followig quadric form Name the quadric x + y + z + 7xz + 5 =
Chapter 5 Groups 5 Some Prelimiaries Defiitio 5 (Operatios o Sets) Uio: Itersectio: Subtractio: A B = A B = A B = { x x A or x B} { x x Aad x B} { x x A x B}, Defiitio 5 A mappig f : A B from a set A to set B is a rule which assigs to each elemet x A a uique elemet f ( x) B Remark 5 A mappig ca be defied i the followig ways: By arrows For example, a b c d represets a mappig from {,,} to {a,b,c,d}
By a two row matrix For example, σ = a a b 4 c 5 d defies a mappig from {,,,4,5} to {a,b,c,d} By formulas For example, f(x)=six defies a mappig from R to R Defiitio 54 Let f : A B ad g : B C be mappigs () If A A, B B, the ad f f { x } ( A ) = f ( x) A {, B } ( B ) = x x A f ( x) () The compositio mappig g o f : A C is defied by ( x) = g( f ( x) ), x A g o f f g A B C Defiitio 55 Let A be a o-empty set ad a o-egative iteger The a biary operatio o o A is a mappig o : A A A A -array operatio o A is a mappig from Cartesia product of umber of copies of A A AL A to A, where A AL A is the Defiitio 56 Let be a biary operatio o a set A () is associative if for ay a, b ad c A, () is commutative if for ay a, b A, ( b c) = ( a b) c a
a b = b a () e A is a idetity if e x = x e = x for all x A (4) If e is a idetity of A, the a is called a iverse of b if a b = b a = e Exercise 5 Let R be the set of all real umbers Defie a b=a+b for all a,b i R (i) Is this operatio associative? (ii) Is there a idetity? Let A={ a, b, c } Defie a operatio o A by the followig table: a b c a a b c b b c a c c a b (i) (ii) (iii) Is this operatio associative? Is there a idetity? Is this operatio comutative? 5 Permutatios Defiitio 5 A permutatio of a set X is a bijectio from X to X, that is a oe-to-oe ad oto mappig Example 5 is a permutatio of the set = {,,} X σ = σ = is a permutatio of the set = {,,,4,5} X 4 5 5 4
Remark 5 () If σ = σ is a permutatio of the set {,,,} the secod row lists all elemets i {,,,} () There are all together! L ( ) ( ) ( ) σ L σ L, where σ (i) deotes the image of i uder σ, the L, ad thus there is o ay repetitio differet permutatios of {,,,} L () The compositio of two permutatios of X is a permutatio of X So the compositio defies a biary operatio o the set of all permutatios of X (4) The idetity mappig : X X is a permutatio of X, where X (a)=a for ay a X X Defiitio 54 The family of all the permutatios of a set X, deoted by S X, is called the symmetric group o X Whe X = {,, L,}, S X is usually deoted by S, ad it is called the symmetric group o letters Theorem 55 Let X be a o-empty set The for ay () σ ( γ o β ) ( σ oγ ) o β o = () X oσ = σ ox = σ for each σ S X σ, β ad γ i S X we have the followig results: () For each σ S X, its iverse mappig σ is also a permutatio of X Ad σ o σ = σ oσ = X I terms of the Defiitio 56, these mea that compositio is associative, X is a idetity ad every oe has a iverse For two permutatios σ ad β of X we simply use αβ to deote their compositio Defiitio 56 Two permutatios σ ad β of a same set X are said to be disjoit if ay elemet that is moved by oe of the permutatios will be fixed by aother, that is for ay x X, σ ( x) x implies β ( x ) = x ad β ( x) x implies σ ( x ) = x Defiitio 57 Let i, i,, i L r be some distict itegers i {,,,} α L If α S fixes the other itegers ad ( i ) = i, α( i ) = i, L, α( ir ) = i the α is called a cycle of legth r, ad is deoted by
( i i L ) i r A cycle ( a a ) of legth is called a traspositio Example 58 () 4 5 6 σ = 4 5 6 is a cycle of legth 4 So σ = ( 4), or = ( 4 ) () 4 5 6 σ = 4 6 5 is ot a cycle σ Exercise 5 Prove if σ ad β are disjoit permutatios i S, the σ β = β σ Prove every σ S ca be writte as a product of some of the followig permutatios Prove for ay cycle ( i ( i ( ), ( ),, ( ) i i k ) ( k i i k ) we have i k i i )=() 5 Eve ad Odd Permutatios Propositio 5 Every permutatio ca be writte as a product of disjoit cycles Example 5 σ = 4 4 5 5 6 = 6 ( )( 4 5)
( )( ) 6 5 4 7 6 5 4 7 6 5 4 = = β Propositio 5 Ay permutatio ca be writte as a product of traspositios Example 54 ( ) ( )( )( )( )( ) 6 5 5 4 4 6 5 4 6 5 4 6 5 4 = = = σ ( )( ) ( )( )( )( )( ) 6 5 5 4 4 6 5 4 4 6 5 6 5 4 = = = β Defiitio 55 S σ is called a eve permutatio if it ca be writte as a product of a eve umber of traspositios Otherwise it is called a odd permutatio Example 56 ( )( ) ( )( )( )( ) 6 5 4 6 5 4 5 6 4 6 5 4 = = = σ So σ is eve ( ) ( )( )( ) 4 4 = = β is odd Exercise 5 Let σ be eve ad β be odd Determie whether each of the followig products is eve or odd: (i) σ β σ ; (ii) β ; (iii) β σ β Prove if σ S commutes with ay β the σ =()
54 Groups Example 54 () Cosider ( R,+), the set R of all real umbers with the additio The (i) x, y R implies x + y R ; (ii) ( x y) + z = x + ( y + z) + ; (iii) there is a zero umber satisfyig x + = + x = x for all x R ; (iv) for each x R, there exists a x R with () I S, (i) σ, τ S implies S (ii) ( τλ) ( στ )λ σ = ; στ ; ( x) = ( x) + = x + x (iii) There exists a elemet ( ) of S with σ ( ) = ( ) σ = σ (iv) For each σ S, there exists σ S with for all σ S ; ( ) σσ = σ σ = () Let A be the set of all the eve permutatios i S The with respect to compositio (4) Let ( R) GL be the set of all real ivertible matrices, the (i) A, B ( R) implies ( R) GL (ii) ( BC) ( AB)C A = ; AB GL ; A has the similar properties (iii) There is a matrix I, which is the idetity matrix, with AI = IA = A for all A GL ( R) ; (iv) For each A ( R), there exists a ( R) GL A GL AA = A A = I with To uify all these examples we itroduce the otio of groups Defiitio 54 A group is a pair (, ) G with G a o-empty set ad a biary operatio o G such that the followig coditios are satisfied: (G) a, b G implies a b G ; (G) a ( b c) = ( a b) c for all a, b, c G ;
(G) there is a idetity elemet e of G such that for all (G4) for each elemet e a = a e = a ; a G there is a iverse a a = a a = e a G a of a satisfyig Example 54 () ( R, + ), (, ),( GL( R), ) * () (, ) S are all groups R is a group, where multiplicatio of umbers Idetity:, Iverse of r : r = r * R is the set of all o-zero real umbers ad is the () Let C ( R) be the set of all cotiuous real fuctios defied o real lie R For ay two f, g C( R) defie f + g : R R by ( f + g)( x) = f ( x) + g( x), x R The ( C ( R ),+ ) is a group Remark 544 (i) If ( G, ) is a group, the the idetity of G is uique (ii) I a group, the iverse of each elemet is uique (iii) I ay group, x a = x b implies a = b Defiitio 545 A group ( G, ) is called a abelia group if the operatio is commutative, ie a b = b a, for all a, b G If G is a abelia group, we usually use + to deote its operatio, use G (or just ) to deote the idetity elemet ad use a to deote the iverse of a G Example 546 () ( R + ), ( C( R), + ), ( Z, + ), are abelia groups () S is ot a abelia group for each But S is abelia (?) () ( ( R ), ) GL is ot abelia for
Exercise 54 Let G be a fiite oempty set ad * be a biary operatio o G such that: (i) * is associative; (ii) for ay x, y, a G, x * a = y * a implies x = y ; (iii) for ay x, y, a G, a * x = a * b implies x = y Prove that (G,*) is a group Prove if G is a group such that a = e for all a G the G is abelia 55 Fiite Groups ad their Operatio Tables Defiitio 55 A group ( G, ) is called a fiite group if G is a fiite set The umber of elemets i a fiite group G is called the order of G, ad is deoted by G Let ( G, ) be a fiite group, where { a, a, } G =, L a We ca use a table to represet the operatio o G, it is called the operatio table of G a a M a a a a a a a M a a a a a a a M a L L L M L a a a a a a M a Example 55 Let {[ ],[ ],[ ] } Z = L Defie [ a ] + [ b] = [ a + b] The ( Z ) is a group [ ] is the idetity The iverse of [ a] is [ a],+ 56 Subgroups
Defiitio 56 Let (, ) G be a group ad let H be a o-empty subset of G H is called a subgroup of G if H is itself a group with respect to the operatio o G Propositio 56 Let (, ) G be a group with a idetity e The a o-empty subset H of G is a subgroup of G if ad oly if H satisfies the followig coditios: (i) a, b H implies a b H ; (ii) a H implies a H H B Example 56 () ( Z,+) is a subgroup of ( R,+) ad ( Q,+) is a subgroup of ( R,+) ratioal umbers () Z = { k k Z} is a subgroup of (,+) Z () H = {[ ],[ ],[ ] } is a subgroup of ( ) 6 6 4 6 (4) A is a subgroup of S Z 6,+, where Q is the set of all Theorem 564 (Lagrage) If H is a subgroup of a fiite group G, the the order of H divides the order of G Example 565 () The order of S 4 is 4! = 4 = 4 If H is a subgroup of S 4, the the order of H must be either, or, or, or 4, or 6, or 8, or, or 4 So the orders of all possible subgroups of S 4 are,,,4,6,8,,4 Z,+ are ad So there are oly two = {,, } Z 5,+?) () The orders of all possible subgroups of ( ) subgroups: H = {[ ] } ad H [ ] [ ] [ ] (What are the subgroups of ( ) () Let a be a elemet of the group ( G, ) The is a subgroup of G Here we assume that a { a } = Z
a ( a ) = e, a = a al a, a = (>) The subgroup a is called the subgroup geerated by the elemet a The order of a elemet a of a group G is defied to be the order of the subgroup a Exercise 56 Let G be a group ad Z(G)={ a G : xa = ax, x G } Show that Z(G) is a subgroup of G This subgroup is called the ceter of G Let A ad B be subgroups of a abelia group G ad let AB={ ab : a A, b B } Show that AB is a subgroup of G 57 Symmetry Groups Much of the importace of groups comes from their coectios with symmetry Just as umbers ca be used to measure size, groups ca be used to measure symmetry With each figure we associate a group, ad this group characterize the symmetry of the figure Defiitio 57 Let R be the Euclidea plae A isometry of the plae is a mappig preserves distace betwee poits p : R R that u T T(v) v T(u) Example 57 () Ay rotatio about a fixed poit i the plae is a isometry () The reflectio of the plae through a lie L is a isometry () Ay traslatio is a isometry
(4) A glide-reflectio is a traslatio i the directio of a lie followed by a reflectio through the lie A glide-reflectio is a isometry Theorem 57 A isometry of the plae is either a rotatio, a traslatio, or a glide-reflectio Defiitio 574 Let T be a plae figure The set of all isometries of the plae that leave T ivariat form a group which is deoted by M ( T ), ad is called the symmetry group of T Remark 575 By the previous result, it follows that if T is a bouded figure i the plae, the every isometry i M ( T ) is either a rotatio or a reflectio Example 576 Determie the symmetry group of a equilateral triagle Solutio Let A, B ad C be the three vertices of the triagle, ο be the cetre of the triagle, ad m AB, m BC ad m CA be the mid poit of AB, BC ad CA respectively The the symmetry group cosists of the followig isometries: = idetity permutatios = rotatio clockwise aroud ο = rotatio 4 clockwise aroud ο 4 = reflectio through lie Am BC 5 = reflectio through lie Bm CA 6 = reflectio through lie Cm AB The operatio table of this group is the followig:
ο 4 5 6 4 5 6 5 6 4 6 4 5 4 4 6 5 5 5 4 6 6 6 5 4