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

  1. Mar 28, 2010 #1
    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)=(k2a,k2b), then V is not a vector space. Find all axioms that fail (and explain why they fail).

    2. Relevant equations

    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

    3. The attempt at a solution
    I couldn't figure out how to start.
    Last edited: Mar 28, 2010
  2. jcsd
  3. Mar 28, 2010 #2
    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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