Vector Spaces and Vector Subspaces. Remarks. Euclidean Space

Transcription

1 Vector Spaces ad Vector Subspaces Remarks Let be a iteger. A -dimesioal vector is a colum of umbers eclosed i brackets. The umbers are called the compoets of the vector. u u u u Euclidea Space I Euclidea space distaces are calculated usig a geeralizatio of the Pythagorea Theorem from two dimesios, called the -orm uv, i i d u v i Spaces usig other defiitios of distace are ofte referred to as No-Euclidea spaces. Other commo orms are the -orm, also called the taicab orm, the p-orm ad the -orm 3

2 Euclidea -space 4 Euclidea -space is the set of all -dimesioal vectors,deoted R :,,, R R Euclidea -space 5 :, is commoly called the Euclidea plae R R Euclidea 3-space :,, is commoly called Euclidea 3-space or simply 3-space R R

3 Remarks May cocepts cocerig vectors i R ca be eteded to other mathematical systems We ca thik of a vector space i geeral, as a collectio of objects that behave as vectors do i R The objects of such sets are also called vectors 7 Eample a b Let M : a, b, c, d R c d a b This matri directly correspods to the vector c d 8 Our Nomeclature K the field of scalars a, b, c, or k the elemets of K V the give vector space u, v, w the elemets of V W a vector subspace space of V for every there eists 9 3

4 Vector Space Let K be a give field ad let V be a oempty set with rules of additio ad scalar multiplicatio which assigs to ay u, v V a sum u v V ad to ay u V ad k K a product ku V We are sayig it is closed uder additio ad scalar multiplicatio The V is called a vector space over K if the followig aioms hold: Vector Space Aioms A For ay vectors u, v, w V, u v w u v w A There is a vector i V, deoted by called the zero vector, for which u u for all uv A3 For each vector uv, there is a vector i V, deoted u, for which u u A4 For ay vectors u, v V, u v v u M For ay scalar k K, ad ay vectors uv, V, k u v ku kv M For ay scalars a, bk, ad ay vector uv, a b u au bu M3 For ay scalars a, bk, ad ay vector uv, ab u a bu Vector Space Aioms M4 For the uit scalar K, ad, u u for ay vector uv 4

5 Eample Let be a iteger ad let P the set of all polyomials of degree at most Polyomial Space Members of P have the form P a a t a t a t where a, a,, a are real umbers ad t is a real variable To prove P is a vector space, we must verify the aioms are true. To verify the aioms we begi with: Let p t a a t a t, q t b b t b t, r t c c t c t, with c ad d scalars 3 Eample Polyomial Space Provig Aiom A p q r is defied as p qt r t t t t t t t t t p q r p q r The polyimial p q r p q r p q r Therefore Provig Aiom A p + p a a t a t pt Hece p + = p t t a a t a t 4 Eample Provig Aiom A3 Polyomial Space q q t t t t b b b b t b b t Hece q q Provig Aiom A4 The polyimial Therefore p q is defied as p qt pt qt p qt a b a b t a b t of degree at most. So p q is i P 5 5

6 Eample Polyomial Space Provig Aiom M p q p c p q cp cq c t c a b c a b t c a b t So ca ca t ca t cb cb t cb t c cq t Provig Aiom M c d p q t c d a c d a t c d a t ca ca t ca t da da t da t Hece c d p cp dp 6 Eample Polyomial Space Provig Aiom M3 p cd t cda cdat cda t dp Hece cd p cdp Provig Aiom M4 c da da t da t c t p t a at a t a at a t p t Hece p p 7 Eample Fuctios Space Let K be a arbitary field ad X be ay oempty set. Let V be the set of all fuctios from X ito K. The sum of ay two fuctios f, g V is the fuctio f g V defied by f g f g X To prove this, oe must verify the aioms A, A, A3, A4, M, M, M3, ad M4 8 6

7 Eample For eample, we will verify A for fuctios To show f g h f g h A For ay vectors u, v, wv, u v w u v w they assig the same value to each X Fuctios Space we must show f g h f g h f g h f g h f g h f g h f g h 9 Remarks Actig o the field of poyomials, differetiatio ad itegratio are trasformatios Eample a which is a equivalet P3 a to writig a3 The derivative is the d p t a a t a t 3 3 dt 3 Polyomials of degree 3 3 p t a a t a t a t 3 3 which is equivalet to writig a a P 3 3a3 we ca write P DP where D is our trasformatio matri 3 3 7

8 Eample Polyomials of degree 3 3 p t a a t a t a t 3 3 To fid the trasformatio matri, we cosider what happes to each uit vector The D 3 3 Eample p t t t t 3 3 P P dp t 3 8t8t 3 dt 3 Subspaces of Vector Spaces Let W be a subset of a vector space over a field K. W is called a subspace of V is W is itself a vector space over K with respect to the operatios of vector additio ad scalar multiplicatio V. W is a subspace of V if ad oly if () W is oempty (or: W) () W is closed uder vector additio u, ww implies u ww (3) W is closed uder scalar multiplicatio v W ad k K implies kv W 4 8

9 Alterate Defiitio W is a subspace of V if ad oly if () W () u, v W ad a, b K implies au bv W It is usually easy to check () of the this defiitio All subspaces pass through the origi To prove, we must verify each aiom To disprove, we eed oly oe couter-eample 5 Remarks A liear subspace is usually called a subspace 6 Subspaces Give a vector space V, the set Z cotaiig oly the zero vector is a subspace of V ad is called the trivial subspace A subspace i R is a lie through the origi A subspace i R is a plae through the origi 7 9

10 Eample a Let H = : a ad b are real b We will verify that H is a subspace of R 3 Let a b is i H, H Addig two vectors i H always produces aother vector whose secod etry is ad therefore H is closed uder additio 8 Eample a Let H = : a ad b are real b Multiplyig a vector i H by a scalar produces aother vector i H. Hece H is closed uder scalar multiplicatio Hece H is a subspace of R 3 Our subspace is a plae i R 3 9 Eample H : is real y Let H H is ot a subspace 3

11 Eample H : is real Also, we could have used this: Let u=, v The u v H 3 Hece H is ot a subspace of R 3 p p Theorem If v, v are i a vector space V, the Spa v, v is a subspace of V To prove a vector is a subspace we use Theorem To prove a vector is ot a subspace we provide a specific eample where oe of the aioms a, b, or c (from the defiitio of a subspace is violated 3 Proof: a. is i Spa v, v, sice = v v v p p p b. To show Spa v, v is closed uder vector p additio we choose two arbitaty vectors i Spa v, v u a v a v a v p p ad v b v b v b v Theorem p p 33

12 u v a v a v a v + b v b v b v p p p p a v b v a v b v a v b v p p p p a b a b a b u v Spav vp u cav av apvp v v v Hece, c. c = ca v ca v ca v p p Hece c u Spa v, v p Theorem p Sice properties a,b,c are satisfied p p p Spa v, v is a vector subspace 34 Eample V a b,a 3 b : a, b R Is V a subspace of R? We write i vectors i colum form a b a b a b a 3b a 3b 3 Let u=, ad = v 3 V=Spa uv, ad therefore V is a subspace of R by Theorem 35 Eample 3 Is H a subspace of R? a b H a : a, b R a a b a a b a H, hece H is ot a subspace 36

13 Eample Is the set H of matrices of the form a b a subspace of M? 3a b 3b a b a b 3a b 3b 3 3 H Spa, 3 3 Hece H is a subspace of M 37 Eample Show C(R) is a subspace Let C(R) be the subset cosistig of cotiuous fuctios, the field R, ad the vector space V be the set R R of all fuctios from R to R We leared i Calculus the sum of cotiuous fuctios is cotiuous We also leared i Calculus the product of a umber ad a cotiuous fuctio is cotiuous Hece C(R) is a subspace 38 3