(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine whether set is a vector space. If not, give at least one axiom that is not satisfied.

the set of vectors <a1,a2>, addition and scalar multiplication defined by:

<a1,a2>+<b1,b2>=<a1+b1+1,a2+b2+1>

k<a1,a2>=<ka1+k-1,ka2+k-1>

3. The attempt at a solution

For Vector Addition:

1) well since a1 and a2 is not restricted, the vector spaces are all real entities in V

2) Rule:x+y=y+x;

<a1,a2>+<b1,b2>=<a1+b1+1,a2+b2+1>=<b1,b2>

3)Rule:(x+y)+z=(x+(y+z);

(<a1,a2>+<b1,b2>)+<0,0>= <a1,a2>+(<b1,b2>+<0,0>)

4)Not sure how to prove is this set has a unique vector O in V such that O+x=x+O

5)Rule: There exist a vector where x+(-x)=(-x)+x=O;

Since the set is not restricted, there exist a negative vector where a1+(-a1)=(-a1)+a1

For Scalar Multiplication:

6)Since the set is no restricted, any scalar would be in the space

7)Rule:k(x+y)=kx+ky;

Not sure how to prove this one....

8)Rule: (k1+k2)x=k1x+k2x

nor this one

9)Rule:k1(k2x)=(k1k2)x

or this one

10)Rule: 1x=x

This one obviously satisfies

Can someone give me hints on the ones i didn't get and also see if the ones i did is correct?

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# Homework Help: Proving If a set is a Vector Free Space

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