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 Proof (rank)

  1. Feb 8, 2010 #1
    1. The problem statement, all variables and given/known data
    A is an c x d matrix. B is a d x k matrix.

    If rank(A) = d and AB = 0, show that B = 0.

    2. Relevant equations

    3. The attempt at a solution
    My textbook has a solution but I don't understand it:

    The rank of A is d, therefore A is not the zero matrix. (I asked my prof why d can't be equal to zero, he said it just couldn't...?)

    If you left multiply A by some elementary matrix to bring it to row echelon form, you get a matrix that looks like:
    [ 1 * * * ... *
    0 1 * * ... *
    0 0 1 * ... *
    0 0 0 0 ... 0] (NOTE: * are arbitrary numbers)

    And we will write B as a column (1 x k), consisting of [B1, ... , Bd]T

    Multiply A and B together, and you get a column that looks like [R1, R2, ... 0, 0, 0]T

    For AB = 0, then Ri = 0. Then since A is not zero, B is 0.

    This proof seems to make no sense. Why are we writing B as 1 x k? It says in the question B is d x k! Also if A is not zero then why can't you say right off the bat that AB = 0 implies B =0?
  2. jcsd
  3. Feb 8, 2010 #2
    because these are matrices not numbers. for example
    Code (Text):

    A= [0 1
       0 0]

    B=[1 0
       0 0]
    AB=0 yet neither A or B are 0.

    as to why they say 'write B as 1xk', maybe they mean write Bv (i.e. B times an arbitrary vector) as a 1xk?
    Last edited: Feb 8, 2010
  4. Feb 8, 2010 #3
    But when A is in row echelon form and you multiply it by some B, the because the solutions are zero the entries of B must be zero??
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook