Results 1 to 3 of 3

Math Help - Basis for Col(A)/Nul(A)

  1. #1
    Newbie
    Joined
    Nov 2006
    Posts
    21

    Basis for Col(A)/Nul(A)

    Given the Matrix A = [[1,3,-4,2,-1,6],[0,0,1,-3,6,0],[0,0,0,1,4,-3],[0,0,0,0,0,0]] (this matrix is 4x6);

    1.) Find a basis for col(A^T)

    2.) Show that nul(A) is orthogonal to col(A^T)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Nov 2006
    Posts
    21
    Ok so my hint for this was:

    For 1.) Reduce (A^T)^T = A to echelon form (for your problem it already
    is in echelon form) then transpose the non-zero rows. Or, reduce A^T to
    echelon form then choose the columns of A^T which contain leading row
    entries (i.e. pivots). There is, in this case, a third way: eyeball
    A^T.

    For 2.) Let w be in null(A) and v be in col(A^T). Then Aw = 0 and
    v = A^Tx for some x. Compute w dot v = w^Tv = ...


    For #1, I found the transpose of A to be:

    [[1,0,0,0],[3,0,0,0],[-4,1,0,0],[2,-3,1,0],[-1,7,4,0],[6,0,-3,0]]

    (6x4 matrix)

    When I rref the transpose of A I get:

    [1,0,0,0], [0,1,0,0], [0,0,1,0], [0,0,0,0], [0,0,0,0], [0,0,0,0]


    The first 3 columns of pivots, if I did that correctly...but whats the basis? The first 3 columns in the orig?

    And for #2

    What's the nul(A) and when I multiply them I should get 0. So I will have some m * 3 matrix times another matrix? Dot product them?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2006
    Posts
    21
    Well I figured it out.

    I'm just confused if col(A^T) is:

    the vectors: [1,3,-4,2,-1,6], [0,0,1,-3,7,0],[0,0,0,1,4,-3] <--- 3 6x1 vectors

    OR: if it is [1,3,-4,2,-1,6], [0,0,1,-3,7,0],[0,0,0,1,4,-3] <-- 3 1x6 vectors...

    For #2 I was able to find the nul(A) and when multiplying each of the vectors from nul(A) to col(A) I got 0, so I am positive I did this right...which leads me to think the 6x1 vectors are right for col(A) but I want to make sure...

    but...col(A^T) is the same as row(A) which makes me think of the 2nd...so im confused.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Basis of ker L --> Basis of vector space?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: September 17th 2011, 09:57 AM
  2. Replies: 4
    Last Post: August 30th 2011, 05:48 PM
  3. Basis and co-ordinates with respect to a basis.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 5th 2010, 08:26 AM
  4. How many different basis does this set contain?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 27th 2010, 10:02 PM
  5. Basis
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 15th 2008, 01:08 PM

Search Tags


/mathhelpforum @mathhelpforum