5.1 - Dr Doom Lin Alg MATH1850/2050 Section 5.1 Page 1 of 6...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Dr. Doom Lin Alg MATH1850/2050 Section 5.1 Page 1 of 6 Chapter 5. General Vector Spaces. Section 5.1: Real Vector Spaces. Definition: Vector Space.. Let V be an arbitrary nonempty set of objects on which two operations are defined, addition and multiplication by scalars (numbers). By addition we mean a rule for associating with each pair of objects u and V in V an object u + V, called the sum of u and V; by scalar multiplication, we mean a rule for associating with each scalar k and each object u in V an object ku, called the scalar multiple of u by k. If the following axioms are satisfied by all objects u, V, W in V and all scalars k and l, then we call V a vector space and we call the objects in V vectors. @If u and V are objects in V, then u + V is in V. CLOSUV-E 0N6“ MQT‘WJ 2. u——v=v+u. 3. u——(v+w)=(u+v)+w. 4. There is an object 0 in V called the zero vector for V, such that 0+u=u+0=u 5. For each u in V, there is an object —u in V, called the negative of 11, such that u + (—u) = (—u) + u = 0. If k is any scalar and u is any object in V, then ku is in V. 095°“ WW9- SCAUW. nun-'53. 7. k(u + V) 2 kn + kv. 8. (k + l)u 2 kn + Zn. 9. k(lu) = (kl)u. 10. 1n = u. Example: Euclidean n-space, R" with the usual addition and scalar mul- tiplication Dr. Doom Lin Alg MATHl850/2050 Section 5.1 Page 2 of 6 Example: Vector Space of 2 X 2 matrices. Show that the set of all 2 X 2 matrices with real entries is a vector space if vector addition is defined to be matrix addition and vector scalar multiplication is defined to be matrix scalar multiplication. To show this, we have to show that each of the vector space axioms (from the previous page) hold for any three 2 X 2 matrices u, V, W, and any scalars k, l. “ (5 \‘A‘ 09. ‘$ELO)J6$ To“ 1. If u an V are objects in V, then u + V is in V. 5»! (AK! \l= M17. - ‘5‘?“ “ml .97?“ d“ “9“ T“ u¢$“&"~"\‘ Uvn um. . Unfit V“ unfi‘ U“. \] 1H6 )3 u ‘k \l = 6 . a U1“? “1" “It 2.u+v=v+u. So u)in 58.3160 AgAgovE. UJ! ‘— U‘“*V" u\1*u“\ = V\\“'uu Vt1*u\1\ -.-_ \l'l'g Wu" Vu (L113 U11— sz+u11 V1.1} u?!- ' “on “on. 3.u+(v+w)=(u+v)+w. worm 52¢) AS «WE 2. vg=[ ] Vll+wl| V“. + w\1.‘\ [ U.“ +( Vu" Mu) “1,4. (“\1+w\1) ‘\ u+(\m~ '— EH " ' ‘3 Vu’m’u Uri-"wit ulna—(013M120 “12*(011fiwul 4. There is an object 0 in V called the zero vector for V, such that 0 + u = u + 0 = u. 0=[°° av. ~ 00 5. For each u in V, there is an object —u in V, called the negative of u, such that u + (—u) = (—u) + u = 0. ‘11-“ " u\‘l. "I—L '= “uh ' “21 Dr. Doom Lin Alg MATH1850/2050 Section 5.1 Page 3 of 6 6. If k is any scalar and u is any object in V, then ku is in V. “‘La Lg‘xku u ku-l" ‘1' u),- 7. k(u+v)=ku+kv. wém L5 6.2 As Aeouz ’ \g Aw sum. U.“ + V“ uVL" “\t‘k - [ k(ua“+uu) filt“ ‘1‘?) a u“ ‘\’ k J“ ‘4 “\L* kv‘t 1— I. kCu1\*U‘L\) \L(_q'nkuib) k U~u+ ‘4V-H \L “11* kul \. (g+2\= \4[ (114* Uh uh.“ V11. 2.”: iétil-LA; . 8. (k+l)u=ku+lu. you Fhfisy‘ «H15 oFfi. 9. k(lu) = (kl)u. 10. 1n = u. Dr. Doom Lin Alg MATH1850/2050 Section 5.1 Page 4 of 6 Example: Vector Space of Functions. Let V be the set of real-valued functions defined on the entire real line (—00, 00). If f = f(a:) and g 2 9(33) are two such functions and k is any real number, define the sum function f + g and the scalar multiple kf, respectively, by (f + g)($) = f (56) + 9(36) and (WW) = left?) In other words, the value of the function f + g at a: is obtained by adding together the values of f and g at 3:. Similarly, the value of kf at a: is k times the value of f at as. This is the usual addition and multiplication by a constant, which you are familiar with. If f and g are vectors in V, then to say that f = g is equivalent to saying that f 2 9(33) for all as. The vector space V is often labeled as F(—oo, 00). Show that V is indeed a vector space. cLosuue afloat. Aoéufon a \r— Q 8.36 ?(-o°,o°) ‘Iueo {+13 4cm»). (501 Foe Ayw x6 It J (Cu) 8. 36:) ALE OEFquIsJ umqu. ’- Q’g*%3(x) =£C¥)-\-a(>¢) is 5696455 & uméiua , 8: +7) él’C’OOIS). “3 ,gz.3 6 $009,...) Tue): {4&0 = 3%: J7 Fm. Am 435. mm) flak») E (BHUCA ,rl-xelfl .m Rifle): {MA—36c) = (ban/gm: gage), Meta. [‘2 {mwa 6 Room) “New &+(3rw)= QQMKA 41¢. snow Aaler, C&+be\)(x)=((-Q+3)¥kyx) c507 (Jitpbwfitx) = (a) + (3mm) = (6.)), (86:) + (Ace) =({m+qbc»o) + W) = Cw(*b)(x) r\' ka) {(Qfi) wyx). eus'venuf a: 259.0 - cue-Sear _. ZLXFO J ,Jxem Cw- CHECK {+2. =2.—\{ =% is. (eh—Mx)‘ (Z*&)C¥3 = {69 ’olxelfl. (but <&\—z. )cx) = {cam—sz) = haw: Qty) =__. -. Cube). en'squ ac Lie-amide ossecr 4%é $000,”) Mean ’gégéoom). -e 563'}. $7 Gmcx3= -£c«). (Ham Tuwr {4r : = 2. @ cmsutu; wot-:0. sum. ,«uu'p: ‘w fiécbmw) 8. [46¢ , saw) H; e FL—p;°°)' (BUT (pg m‘o 37 (Why): L266) So [44 Fem”), [C Q ,3 e FC“'°N°) 2: k6 sumo = A29. saw) (’Q¥=a.))bc) = Q‘Q—HLSDGO) ’J' xélk. (501' (lg (&+-b))(,x) = (x) = 604-38)) = \L«<(z) 4: ‘4809 = (kQBCK) 4%)(9 = M 3. \«QM. ‘lcu FI'UCSH Tue 2,651: Dr. Doom Lin Alg MATHl850/2050 Section 5.1 Page 5 of 6 (A) Example: 6 page 225 from text. Let V be any plane through the origin in R3. This example shows that the points in V form a vector space under the standard addition and scalar multiplication operations for vectors in R3. @ Example: Let V be the line y = a: + 1 in R2. Define addition and scalar multiplication as the standard addition and scalar multiplication of vectors in R2. Show that the points in V do not form a vector space. F“ G) We“ 0”“ “EW- @ 4® (WE OWE“- % Mons A¢E Goa-Gina) mm W} (SEm'c A vac-rm. some). SJ” g1cu‘au1-Nfi) 2‘ 2=(\’\;\’1.V~s) Am 1» #1 “AH: qx -\-lo‘3 *0; =0 so Au\+bu1-CU‘3=O 5‘ QVVEEV'L‘LCUb-ao. pm) 95+! —. (“fl—u“ ,utwa, ,Uks-Ns) & 0~(U~.W\)+S(u;w,,)4 C(u..,w«..\= ans-‘9 Lit-\Cuz, + qvfl‘odficv3 = O ‘\' O '3 0. So (45w 6 @ 8A1 u A5 meme 8. l; Any <(ALA9. Ckéll) \éc; : (V‘M )\4“1)\‘U‘5) afiub+$<kua+cflcufi= \L (Qu.+lOU~-L-L Cu!) = L‘O " O . so lug. G so \I (s A vamp, $0Acc. ' ‘ \1 F09. @ : Cuecv. @ 1F 15 ZgéU (on LNG B‘XJM), mace. u: (£4! e 30 u=(u‘,u¢) ’ um: LU“ A ytcv|)u1\w\1\\' V-LlVl'Vl .— ufiul: Cu‘+l)r(\1v\-\)= (\L‘+V\\‘\'2. f (u.\-\-\J\\-\'\ 5., guy C—\} SO x] is Not A decwt sax-€- ’— Dr. Doom Lin Alg MATHl850/2050 Section 5.1 Page 6 of 6 The Zero Vector Space. Let V consist of a single object, which we Will denote by 0, and define addition as 0+0=0 and scalar multiplication as k0 = 0 for all scalars k. It is easy to check that all the vector space axioms are satisfied. We call this the zero vector space. Theorem 5.1.1: Properties of Vector Spaces. Let V be a vector space, 11 be a vector in V, and k a scalar; then: (a) 0n = 0. (b) k0 = 0. (c) (—1)u = —u. (d) Ifku=0,thenk=00ru=0. @ 0 g [email protected])g = Qg 1:02. émce OgéU J l1's ween-(us EXIST: -Ot3 $9 Og+C—Og):OgLOg+C-'Og) So Q=Og+Q , Hence avg ® \LQ -.-. \c(O_iQ).—_ \ggrkg AM ~—lLQ To earn sides, a(,ET lLQ=Q. Q GI)“.\.g—;(—\)B.\. s (—l-Eog =0té=g Agaat E*(_\)%=g ASWELL- (1) 3w bar-Q JUL-#0 Cum“ Saw-Q) $0 téll ...
View Full Document

This note was uploaded on 11/10/2009 for the course MATH 1850 taught by Professor Mihaibeligan during the Spring '09 term at UOIT.

Page1 / 7

5.1 - Dr Doom Lin Alg MATH1850/2050 Section 5.1 Page 1 of 6...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online