1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook