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Homework Help: Howto define the Column space of nxn matrix

  1. Feb 8, 2010 #1
    1. The problem statement, all variables and given/known data

    I thought that if you have a square matrix then the column space is the set of all linear independent vectors which can be written as a linear combinations of the others? Which inturn is the same as range of the Matrix?
    Am I wrong?
     
  2. jcsd
  3. Feb 8, 2010 #2
    The column space of a matrix is the set of all possible linear combinations of its column vectors.
    It is the same as the range for the corresponding transformation matrix.
     
  4. Feb 8, 2010 #3
    and thats the same as the range of the matrix?
     
  5. Feb 8, 2010 #4
    Yes, it is the same as the range of the matrix (for the corresponding transformation).
     
  6. Feb 8, 2010 #5
    It little follow-up question the same n x n matrix B is a 6 x 6 then then following how do I show if the matrix equation Bx = c has a solution for all [tex]c \in \mathbb{R}^6[/tex] is that if all the columns in B are linear independent and thusly spans the whole of [tex] \mathbb{R}^6[/tex] and then claim is true? and if they don't the claim is false for the matrix B?
     
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