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

If we define V as the set of vectors in R^2 with vector addition defined as it normally is, but scalar multiplication defined to be k(a,b)=(k^{2}a,k^{2}b), then V is not a vector space. Find all axioms that fail (and explain why they fail).

2. Relevant equations

Axioms:

1. Ifuandvare objects in V, thenu+vis in V.

2.u+v=v+u

3.u+(v+w)=(u+v)+w

4. There is an object0in V, called azero vectorfor V, such that0+u=u+0=ufor alluin V.

5. For eachuin V, there is an object -uin V, called anegativeofu, such thatu+(-u)=(-u)+u=0.

6. Ifkis any scalar anduis any object in V, then kuis in V.

7.k(u+v)=ku+kv

8. (k+l)u=ku+lu

9.k(lu)=(kl)u

10. 1u=u

3. The attempt at a solution

I couldn't figure out how to start.

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# Homework Help: Linear Algebra: Find all axioms that fail k(a,b)=(ak^2,bk^2).

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