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How can I tell if this is a vector space?

  1. Oct 28, 2007 #1
    1. The problem statement, all variables and given/known data
    A set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold.

    (x,y,z) + (x',y',z') = (x+x',y+y',z+z') and k(x,y,z) = (kx,y,z)

    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 = for all un 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 + m)u = ku + mu

    9. k(mu) = (km)(u)

    10. 1u=u

    3. The attempt at a solution

    I have no idea what they are asking for, the back of the books say it fails one axiom which is: (k+m)u = ku +mu Which is axiom 8 in my book.
    Last edited: Oct 28, 2007
  2. jcsd
  3. Oct 28, 2007 #2
    k(x,y,z) = (kx,y,z)

    notice that scalar multiplication only goes to the 1st coordinate.
  4. Oct 28, 2007 #3
    I know that!! I'm taking linear algebra, I should be able to know that atleast! read my question again, tell me why, in good mathematical proof that axiom 8 does not apply. Thats all I need. look at axiom 8 please!!!
  5. Oct 29, 2007 #4


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    Staff Emeritus
    Science Advisor

    They are asking you to check each of the axioms to see if they apply.

    Since the addition defined is exactly the usual addition in R3 which you know is a vector space, you know that all axioms involving only addition of vectors,(1- 5), apply. Look at what axiom 8 says: (k+m)(x,y,z)= ((k+m)x,y,z)= (kx+ mx,y,z). But k(x,y,z)+ m(x,y,z)= (kx,y,z)+ (mx,y,z)= ((k+m)x, 2y, 2z). Not at all the same.
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