1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Basis Proof

  1. May 5, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove (u,v,u+v) can not be a basis for <u,v,u+v>.

    2. Relevant equations

    3. The attempt at a solution

    Let αu+βv+γ(u+v)=0

    Since α/γ,β/γ,1 are not all zeros, therefore, (u,v,u+v) are linearly dependent. Hence it doesn't form basis for <u,v,u+v>.

    Let me know if this is the right approach towards the proof.
  2. jcsd
  3. May 5, 2009 #2
    Showing that (u,v,u+v) are linearly dependent is the correct way to prove they don't form a basis. To show that they are linearly dependent, you only have to find an example of α, β, and γ (not all zero) for which αu+βv+γ(u+v)=0. I would just write down a specific choice of numbers that works; this is not difficult to do by guess and check.
  4. May 5, 2009 #3
    So I guess we can achieve that by saying the following:

    -1*(u) + (-1)*v + 1*(u+v) = 0

    Since none of the constants are actually zero. Therefore, (u,v,u+v) are infact linealy dependent.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Basis Proof Date
Linear Combination Proof of Orthonormal basis Oct 11, 2015
Basis Proof Oct 7, 2012
Vector space basis proof Dec 1, 2011
Elementary Linear Algebra Proof Basis/Dimension Oct 21, 2011
Basis and Determinant Proof Jul 12, 2011