4.1 - Dr. Doom Lin Alg MATH1850/2050 Section 4.1 Page 1 of...

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Unformatted text preview: Dr. Doom Lin Alg MATH1850/2050 Section 4.1 Page 1 of 8 Chapter 4 - Euclidean Vector Spaces Section 4.1 - Euclidean n-Space Definition If n is a positive integer, then an ordered n-tuple is a se- quence of n real numbers (a1, a2, . . . ,an). The set of all ordered n-tuples is called n-space and is denoted by R". Examples: 1- so,“ , \Q:L Au oufiuan PAD. is (—59.) cm Tuiuv. 0(— (431.) A5 A Pain“: um“ x—coq‘p -I\) Teen”: 1: on, As A occur. 1\-\&1’ Couums we: 0mm (0,9) To THE «im- #9 amass 3—1091}: (“4,79 Dr. Doom Lin Alg MATHl850/2050 Section 4.1 Page 2 of 8 Definitions Two vectors u = (U1,U2,...,Un) and V = (v1,v2,...,vn) in R" are called equal if U1=U1, UQZUQ, ..., unzvn. The sum u+V is defined by u+v= (u1+v1,uQ+v2,...,un—i—vn), and if k is any scalar, the scalar multiple ku is defined by [€11 2 (km, ICUQ, . . . , The operations above are called the standard operations on R". The negative of u, denoted by —u is the vector —11 = (—U1,—U2,. . . , —Un) and the difference of vectors is defined by V—uzv+(—u) Examples: 1. if(a,3,b,0)=(1,3,a,0) then Q=4 , Lam—.4 2. (2,1,—1,3,5)+(6,1,1,0,—3) = (9.,1,o,3,z) 3. 4(1,7r,2,0) = (HAMLO) Dr. Doom Lin Alg MATH1850/2050 Section 4.1 Page 3 of 8 Theorem 4.1.1 Properties of Vectors in R”. If u =(u1,u2,...,un),v = (v1,v2,...,vn) and W = (w1,w2,...,wn) are vec- tors in R" and k and m are scalars, then: (a) u __ V = V + u COMMUTNYRhTV b) u —— (V + w) = (u + v) + w Maw/xiiqu orucm {a 10;“ is (o,e,_,,,o) LBS = \L Chg = ¥(u‘wuu1+v1 ’ . . _, Low“) = (kLuwu‘), \L (Mm ,waM) = (item—kw , \cuu huh-.. , hum-kw) = (Mu, ’ kq.“ ...)kuq+(kv.,kvh...,kuh) = ‘4 g -\ k! = REES Definition If u = (ul,u2, . . . ,un) and V = (111,112, . . . ,vn) are any vectors in R", then the Euclidean inner product 11 ~ V is defined by u~v=u1v1+uQv2+...—i—unvn Example: (2, —1,0,5) ~ (1,3,5, —2) = 1-H L—n-H o-S +SGI)= -u,. Dr. Doom Lin Alg NIATHléfioU/QUSU Section 4.1 Page 4 of 8 Theorem 4.1.2 Properties of Euclidean Inner Product If u,v and W are vectors in R” and k is any scalar, then: u - v = v - u coMMUTATEUi’VY (b) (u+v) -w=u-w+v-w (c) (ku) - v = Mu - v) (d) v - v 2 0. Moreover, v - v = 0 if and only if v = 0. 1 _ . Proof: (51.). V41 —. vf+vl+ ...+ U“ >/O since we"! mom mvucemwc W5. 1. . 1. 1. g 7- mLeoum vf+~f1}*-—-Nh ‘0 ‘« UL=V1.‘-~‘IM=O. \J‘:\I,L=“_:\lh=0 Example: With u = (3,—2,1,2),v = (2,0? —1} —1),w =(1,1,—1,1) LEE: (US-Ely t (I; lhz’o’llaojll-‘W? LH$=£H$ ‘h—lis is Acrr AWOOC. Lake-v3 +1.51 =1+l= Li Definition we define the Euclidean norm (or Euclidean length) of a vector u 2 (U1, U2, . . . ,un) in R" by: ||u|| = (11‘ 1101/2 = lu}«u¢1’v.-.+ul' and we define the Euclidean distance between points u 2 (U1, U2, . . . ,un) and V = (111,112, . . . ,vn) in R" by d(u, v) = ||ul — VI I-= tecvo‘arcuz-vm “scan-um Examples: 1. ifu= (2,3,—1,1) then = Lu‘flH-H =55 2. with u as above and V = (1, —2,3,2), d(u,v) = ||u — V|| =\l( l JSF‘D'D“ =m’ =§L€5l Dr. Doom Lin Alg MATH1850/2050 Section 4.1 Page 5 of 8 Theorem 4.1.3 Cauchy-Schwarz Inequality in R”. If u = (U1,U2, . . . ,un) and V = (111,112, . . . ,vn) are vectors in R", then / DES-M16 WT? \MLU“ lu'V\ SHUIIIIVII Example: With 11 = (2, —1,2),V = (—1,3,0) 1w: -9 , \2‘2\=6 \lgl\= ‘Sqmu =3 , “gr—Sew ’ @ \guxfl-«S 4 “T = “\MW“ Theorem 4.1.4 Properties of Length in R”. If u = (U1,U2, . . . ,un) and V = (111,112, . . . ,vn) are vectors in R", then: (a) ||uH 2 0 “truce-MM» (b) :0 iffuz 0 EC) ||k11||= lkl ||11|| d) I|u+ VH g + (Triangle Inequality) Pmofia.) \\u\\=0 1% who «1 typo neg wQ Cam m) (A) llgwui: (Jun-(WNW kyle-wry =t\-\l +1wg+tyn W Qm‘f—Scumz Dewy aka-gxezuguugt m M1 .m}: :—————‘ ) w" 3 < ugu‘muguugutgu“ =(\u,u\+\\x_n\)z a So have (llgHuyuSL , Sp llujgté ugu+\\\_2_\\ Dr. Doom Lin Alg MATH1850/2050 Section 4.1 Page 6 of 8 Theorem 4.1.5 Properties of Distance in R”. If u = (“171% - - - vun)aV = (7117712, - - - 77177,) and W = (101,102, . . . ,wn) are vec- tors in R", then: (a) d(u,v) 2 0 W“. swank {\g—guv/o (b) d(u,v) : 0 iff u = V (c) d(u, V) = d(v, u) (d) d(u,v) g d(u, W) + d(w,v) (Triangle Inequality) Proof: i\u_—!“=OL=> =0 4*) U.=\.I, —— .— (ol) out”): \\\L~‘yk\= Massey-M é [\kyQun-gu = ngMHMwM Theorem 4.1.61fu 2 (U1, ug, . . . ,un) and V = (111,112, . . . ,vn) are vectors in R" with the Euclidean inner product, then 1 1 M = leu+vll2 — EMU—VH2 Proof: 1 2' fly!“ = GAMES-Lqu lglC‘Jclg-y + Ml “\A’Qli" = (erg-(£51) = mu; 9,! + m1 svw'“ \\ WW“ My!“ = Li we (£3 = {—uyguk—‘q [UL—VJ". Dr. Doom Lin Alg MATH1850/2050 Section 4.1 Page 7 of 8 Definition Two vectors u and V in R” are called orthogonal if 11 ~ V = 0. mix: 9&1 g . Examples: With 11 = (2, —1,2,1),V = (1,1,—1,1),w = (1,1,3,1) E R4 \_l~_-'\J='L-\-L-\-\=O so 5.1.;1 'gel-HLi—b‘bfo so gin.) IF V.\y=\-\\*7r\’1=0 5° glw Theorem 3.3.3 If u and a are vectors in R2 or R3 and if a 55 0, then A ‘ a W“ 3 gm: projau 2 H111" a is the vector component of 11 along a _\ u—projau is the vector component of u orthogonal to a pfiai‘. Q Example: With 11 = (2, —1,3),a = (1,0,2) 11-3.: 94-9-9. , [14!]; {S— N A 4'3 I Mil. —.-. = .3—C\’O,1):(%,QHL§6.) All" S WM?“(1"‘233’(%»0a%)=(~v Theorem 4.1.7 Pythagorean Theorem in R” If u and V are orthogonal vectors in R" with the Euclidean inner product, then ||Ul+V||2 = ||u||2+ ||V||2 Proof: llu+\_/\\1'= Lauri-(«tu \lwi‘Jrzkr-gwwiga infirm" _ W =0 (55-! A v Dr. Doom Lin Alg MATH1850/2050 Section 4.1 Page 8 of 8 A Matrix Formula for the Dot Product If we use column notation for vectors U1 711 U2 U2 11 = ' ,V 2 um 7177 and we omit the brackets on 1 X 1 matrices, then MAsz Mecca LL: yTB 7 [VI V1. —-- V'A] (f1 °(_\I\u|+ V‘Lu1*"-+Vv\u“’\=Ly.gi-S= %-\l = t Um [5m 1% Mars. IfAis annxnmatrix, then we get lAu~V=u~ATV u~AV=ATu~V A‘é‘! = \JT(ALQ=@A)% = (MW; = 8%?! Example: Verifying thatIAu ~ V = 11 ~ ATV j 1 2 3 2 1 A: —131,u= 0 ,V= 2 0 0 5 6 1' LHS Aqv- 1° ‘I T L " — ___- q.z=5gy QHS=Q~‘.A\l-'o g=S%/. 3O ‘ C (o ...
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