[理工] [線代]-五題Vector Space證明題
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?
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.
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.
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?
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.
PS：
Axioms:
A1. x + y = y + x for any x and y in V.
A2. (x + y) + z = x + (y + z) for any x,y,z in V.
A3. There exists an element 0 in V such that x + 0 = x for each x in the set V.
A4. For each x in the set V, there exists an element -x in V such that x + (-x) = 0.
A5. alpha(x + y) = alpha*x + alpha*y for each scalar alpha and any x and y in V.
A6. (alpha + beta)x = alpha*x + beta*x for any scalars alpha and beta and any x that belongs to the set V.
A7. (alpha*beta)x = alpha(beta*x) for any scalars alpha and beta and any x that belongs to the set V.
A8. 1*x = x for all x in V.
以上，麻煩好心的大大！（鞠躬
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