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!

Proving that Columns are Linearly Dependent

  1. Mar 16, 2015 #1

    B18

    User Avatar

    1. The problem statement, all variables and given/known data
    Let A be an m x n matrix with m<n. Prove that the columns of A are linearly dependent.

    2. Relevant equations
    Its obvious that for the columns to be linearly dependent they must form a determinate that is equal to 0, or if one of the column vectors can be represented by a linear combination of the other vectors.

    3. The attempt at a solution
    It seems like there has to be more shown to prove this statement, however this is what I have right now:
    Let A be an m x n matrix, and let m < n.
    Then the set of n column vectors of A are in Rm and must be linearly dependent.

    Is this it? or do I need to state a theorem in here somewhere?
     
  2. jcsd
  3. Mar 16, 2015 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Hint: what is the maximum dimensionality of the space spanned by the columns (regarded as column vectors)?
     
  4. Mar 16, 2015 #3

    B18

    User Avatar

    The maximum dimension of the space spanned would have to be m+1, correct? For example if the vectors were from R3 we would need 4 column vectors so that they were linearly dependent.
     
    Last edited: Mar 16, 2015
  5. Mar 16, 2015 #4
    Show that the reduced row echelon form of the mxn matrix will have at most m pivots. Then there are n-m columns without pivots, which can all be expressed as linear combinations of the columns with pivots. Since they are linear combinations of other columns, they are linearly dependent.
     
  6. Mar 17, 2015 #5

    B18

    User Avatar

    It would have been helpful if our professor explained what pivots were. Thanks though Izzy I'll make sense of what you explained and go from there.
     
  7. Mar 17, 2015 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Are you telling me that you think 4 or more 3-component vectors can possibly span a space of dimension 4?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Proving that Columns are Linearly Dependent
  1. Linearly dependent (Replies: 4)

  2. Linearly Dependence (Replies: 7)

  3. Linearly dependent (Replies: 2)

  4. Linear dependance (Replies: 7)

Loading...