1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Linear Algebra: Spans and Dimensions

  1. Feb 25, 2013 #1
    1. The problem statement, all variables and given/known data
    Given v1, v2 ... vk and v, let U = span {v1, v2 ... vk} and W = span {v1, v2 ... vk, v}. Show that either dim W = dim U or dim W = 1 + dim U.

    3. The attempt at a solution
    I'm not really sure where to start. If I knew that {v1, v2 ... vk} was linearly independent, then it would be easy to prove. But I'm not given anything about linear independence so I'm at a loss.
  2. jcsd
  3. Feb 25, 2013 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    What if you reduce ##\{v_1,v_2,...,v_k\}## to a basis set for U to work with?
  4. Feb 25, 2013 #3
    Okay, so if I reduced the generating set for U to a basis, I would know its linearly independent, but then where would I go with it?
  5. Feb 25, 2013 #4
    Do the same for W. Note, W is also spanned by v which may or may not be represented as a linear combination of the other vectors (this is why there are two possible cases for the dimensions).
  6. Feb 25, 2013 #5


    User Avatar
    Science Advisor

    Consider two cases:
    1) v is in U.
    2) v is not in U.
  7. Feb 25, 2013 #6
    Ok, so assuming we reduce the set in U to a basis: then if v is in U the given generating set for W is linearly dependent. Then v would be removed from this set, making it linearly independent. This implies that U and V share a basis, and thus their dimensions are the same. If v is not in U, then the given generating set is linearly independent, and thus is a basis for W. Therefore dim W = dim U + 1. Does this make sense?
  8. Feb 26, 2013 #7


    User Avatar
    Science Advisor

    I would think it was obvious that if v is in U then adding v to U doesn't change the span: span(U)= span(V) so their dimensions are the same. It does NOT follow that "If v is not in U, then the given generating set is linearly independent" but it is not necessary. What ever the dimension of U, if v is not in U, adding it increases the dimension by 1.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted