※ 引述《ruby791104 (阿年：）)》之銘言： : 1.Let S be the set of all ordered pairs of real numbers. Define scalar : multiplication and addition on S by : α(x1,x2) = (αx1,αx2) : (x1,x2)⊕(y1,y2) = (x1+y1,0) : We use the symbol ⊕ to denote the addition operation for this system to : avoid confusion with the usual addition x + y of row vectors. Show that S, : with the ordinary scalar multiplication and addition operation ⊕, is not a : vector space. Which of the eight axioms fail to hold? Sol: u=(x1,x2) u+0=(x1,0) =/=u it's not vector space : 2.Let V be the set of all ordered pairs of real numbers with the ordinary : defined by : (x1,x2) + (y1,y2) = (x1+y1,x2+y2) : and scalar multiplication defined by : α。(x1,x2) = (αx1,x2) : The scalar multiplication for this system is defined in an unusual way, and : consequently we use the symbol 。 to avoid confusion with the ordinary : scalar multiplication of row vectors. Is V a vector space with these : operations? Justify your answer. Sol: u=(x1,x2) 1*u=/=u u+(-u)=(0,2x2)=/=0 it's not vector space : 3.Let R denote the set of real numbers. Define scalar multiplication by : αx = α．x (the usual multiplication of real numbers) : Is R a vector space with these operations? Prove your answer. 這定義一般乘法規則,應該是吧XD 要證就要把八項公理全驗證一次=.=a : 4.Let Z denote the set of all integers with addition defined in the usual way : and define the scalar multiplication, denoted。, by : α。k = [[α]]．k for all k€z : where [[α]] denotes the greatest integer less than or equal to α. : For example, : 2.25。4 = [[2.25]]．4 = 2．4 = 8 : Show that Z, together with these operations, is not a vector space. Which : axioms fail to hold? sol:(1.7 + 2.4)。u = 4u=/=(1.7)。u + (2.4)。u=3u : 5.Let S denote the set of all infinite sequences of real numbers with scalars : multiplication and addition defined by : α｛an} = {αan｝ : {an} + {bn} = {an+bn} : Show that S is a vector space. 自己把8個公理驗證完......... ps:線代證明我不熟,有錯鞭小力一點>"<,有請高手補充m(_ _)m -- 為者常成.行者常至 --

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