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!

Proof: Max number of Linearly Independent Vectors

  1. Sep 13, 2015 #1
    1. The problem statement, all variables and given/known data
    Prove that a set of linearly independent vectors in Rn can have maximum n elements.

    So how would you prove that the maximum number of independent vectors in Rn is n?


    I can understand why in my head but not sure how to give a mathematical proof. I understand it in terms of the number of independent vectors being equal to the rank of the matrix they create and obviously a matrix of dimension n can only have max n pivots. But I don't think that's really sufficient for a proof.
     
    Last edited: Sep 13, 2015
  2. jcsd
  3. Sep 13, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    If you write it down properly, that should be fine.
    The main point: a n x m matrix (for your m vectors) cannot have rank larger than n.
     
  4. Sep 13, 2015 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Yes, it is (with a bit of tweaking). If vectors ##v_1, v_2, \ldots, v_n, v_{n+1} \in R^n## are linearly independent, we should not be able to find ##(c_1, c_2, \ldots, c_{n+1}) \neq(0,0, \ldots, 0)## giving ##c_1 v_2 + c_2 v_2 + \cdots + c_n v_n + c_{n+1} v_{n+1} = \vec{0}##. Assuming, instead, that you CAN find such ##c_i##, you should be able to get a contradiction.
     
    Last edited by a moderator: Sep 13, 2015
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted