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

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Homework Statement



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)=(k2a,k2b), then V is not a vector space. Find all axioms that fail (and explain why they fail).

Homework Equations



Axioms:
1. If u and v are objects in V, then u+v is in V.
2. u+v = v+u
3. u+(v+w)=(u+v)+w
4. There is an object 0 in V, called a zero vector for V, such that 0+u=u+0=u for all u in V.
5. For each u in V, there is an object -u in V, called a negative of u, such that u+(-u)=(-u)+u=0.
6. If k is any scalar and u is any object in V, then ku is in V.
7. k(u+v)=ku+kv
8. (k+l)u=ku+lu
9. k(lu)=(kl)u
10. 1u=u

The Attempt at a Solution


I couldn't figure out how to start.
 
Last edited:

Answers and Replies

  • #2
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If addition is defined as normal, and scalar multiplication is altered, that gives you a hint on what axioms to check.
 

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