Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Expanding linear independent vectors

  1. Sep 11, 2016 #1

    joshmccraney

    User Avatar
    Gold Member

    Hi PF!

    The other day in class my professor mentioned something about expanding linear independent vectors, but he did not elaborate. From what I understand, if ##x_1,...,x_k## are linearly independent vectors in ##V##, where ##dimV=n>k##, how would you extend ##x_1,...x_k## to a basis ##\{ x_1,...,x_n \}##. Lets say ##\{ y_1,...,y_n \}## is a basis for ##V##. By extending the ##x## vectors, do you think he was just referring to including all the ##y## vectors in the set of ##x## vectors that are linearly independent of the ##x## vectors?

    Thanks!
     
  2. jcsd
  3. Sep 11, 2016 #2

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    If you have a set of linearly independent vectors, you can expand that set into a basis simply be adding more vectors. In your example, you can choose ##x_{k+1}## as any vector in V that is not in the span of ##\{x_1, \dots x_k \}##.

    And then, any vector ##x_{k+2}## that is not in the span of ##\{x_1, \dots x_{k+1} \}##.

    Until you have a basis of ##n## linearly independent vectors.
     
  4. Sep 11, 2016 #3

    joshmccraney

    User Avatar
    Gold Member

    OK cool, that's what I thought but I wanted someone else's perspective! Thanks PeroK!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted