Re: [理工] [線代]-五題Vector Space證明題
※ 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
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