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: Proving row space column space

  1. Jun 21, 2011 #1
    A , B are nXn matrices
    t-is transpose
    prove that the space spanned by A's row equals the space spanned by A's columns
    i know that there dimentions are equals
    so in order to prove equality i need to prove that one is a part of the other
    how to do it?

    each column i of (AB)_i=A*B_i
    i was told by my proff that that column i of AB is a member from the span of the columns of A

    but i dont get this result
    suppose the member of B_i column is (c1,c2,..,cn)
    so the multiplication of A by the B_i column
    we get then the first member is dot product from the first row with (c1,c2,..,cn)
    i cant see how its a variation from the A columns?
    Last edited: Jun 21, 2011
  2. jcsd
  3. Jun 21, 2011 #2


    User Avatar
    Homework Helper

    yeah i think you're on the right track,
    A.B = A^T

    now consider a the kth column B, the vector B_k, which when multiplied with A yields the kth column of A_T, (A^T)_k
    A.B_k = (A^T)_k

    so the kth column of A^T is a linear combination of the columns of A, given by the components of B_k.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook