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!

Vector space

  1. Oct 7, 2007 #1
    15. Determine wheter the set is a vector space.
    The set of all fifth-degree polynomials with the standard operations.
    1.u+v is in V
    6. cu is in V

    the back of my book says that axioms 1,4,5, and 6 fail. I dont know why 4,5,and 6 fail. Can anyone help me?
    Last edited: Oct 7, 2007
  2. jcsd
  3. Oct 7, 2007 #2
    hm..if it fails 1 and 6 shouldn't it also fail 7? o.o...Sorry I can't see a counterexample that 1/4/5/6 fails.

    say you have
    where a,b,...,l are arbitrary constants where at least 1 of a,b,...,f and g,h,...,l doesn't equal 0.


    you can write it them out and I think they do hold, or can you give 1 counter-example that proves it doesn't?
    Last edited: Oct 8, 2007
  4. Oct 8, 2007 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Oh boy is that too complicated. 4,5,6 simply fail because 0 is not a 5th degree polynomial, thus there is no 0 to add to u in 4, no 0 for u+(-u) to equal, and 0*u is not a 5th degree poly
  5. Oct 8, 2007 #4


    User Avatar
    Staff Emeritus
    Science Advisor

    Notice that the set of all polynomials with degree less than or equal to 5 is a vector space. But here, you must have polynomials of precisely degree 5. The "0" polynomial is not in that set.
  6. Oct 8, 2007 #5
    to matts response:
    im still getting use to this abstract way of thinking and am not sure if this is stupid or not. For axiom 4 to pass, does the zero vector have to be a 5th degree polynomial? cant the zero vector just be 0? b/c lets say (x^5+x)+(0)=(x^5+x). Whats wrong with the way im approaching this?
  7. Oct 8, 2007 #6


    User Avatar
    Homework Helper

    Well, first of all, you wrote the axioms in such an unprecise manner, that some of them represent nothing. For example, axiom 4 sould be something like: "There exists 0 in V, such that for every x from V, 0 + x = x + 0 = x", etc, which means that merely by reading the axioms you can get the answers to your questions.

    You can't add a polynomial and a number. Of course 0 has to be a polynomial. And since it is not a polynomial of degree 4 (or whatever you defined the space to be, I don't remember right now), it fails to be in the vector space at all.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Vector space
  1. Vector spaces (Replies: 10)

  2. Vector space (Replies: 16)

  3. Vector space (Replies: 2)

  4. Vector spaces (Replies: 6)

  5. Vector space (Replies: 4)