1. Not finding help here? Sign up for a free 30min 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!

Linear Algebra: Vector space axioms

  1. Dec 13, 2006 #1
    1. The problem statement, all variables and given/known data
    One of the fundamental axioms that must hold true for a set of elements to be considered a vector space is as follows:
    1*x = x

    I was given a particular space: The set of all polynomials of degree greater than or equal to three, and zero, and asked to evaluate whether or not it was a vector space or not. The one that doesn't hold true is 1*x=x (according to the solutions), but I don't understand why. I can't find a situation where the above axiom holds true. Could anyone help me out?

    Thanks
    Preet
     
  2. jcsd
  3. Dec 13, 2006 #2

    StatusX

    User Avatar
    Homework Helper

    I'd check the solutions again. The problem is closure under vector addition.
     
  4. Dec 14, 2006 #3

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    The set of vectors and the vector addition must form an abelian group. That is true for those polynomials. However, your polynomial space lacks the "null/zero vector"; since we know that the scalar zero times any vector from your set should give the zero vector. Since the zero vector is not an element of the space, it follows that the polynomials' space plus vector addition plus scalar multiplication do not yield a vector space.

    Daniel.
     
  5. Dec 14, 2006 #4

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    Daniel, the original post says that zero is an element of the set. The problem is indeed vector addition, since [itex]x^3 + (-x^3 + 1) = 1[/itex], for exampl.
     
  6. Dec 14, 2006 #5

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Yes, of course. :redface:

    Daniel.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Linear Algebra: Vector space axioms
Loading...