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October 9th, 2015, 11:18 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,621 Thanks: 117  Column Rank equals Row Rank
Consider Ax=0 $\displaystyle \begin{vmatrix} a11\\ a21\\ a31\\ a41\\ a51 \end{vmatrix}x1+\begin{vmatrix} a12\\ a22\\ a32\\ a42\\ a52 \end{vmatrix}x2+\begin{vmatrix} a13\\ a23\\ a33\\ a43\\ a53 \end{vmatrix}x3=0$ Assume the first two rows are linearly independent (row rank=2). If x1, x2, x3 satisfy first two rows after an elementary row operation on them, they will satisfy them before, hence satisfy all rows. With first two rows in reduced row echelon form only two columns are LI, hence column rank = row rank = 2. For example: x1a1+x2a2+x3a3=0 has a solution, but x1a1+x2a2=0 > x1=x2=0. 
October 12th, 2015, 07:22 AM  #2 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,621 Thanks: 117  Column Rank = Row Rank and Reduced Row Echelon Form Unique
Ax=0 can be written a1x1 +a2x2+...anxn=0 An elementary row operation doesn't change column rank because it doesn't change the solution. So if aixl+amxm+anxn=0 > xl=xm=xn=0 for A in reduced row echelon form, then it is also true for A in original form. In reduced row echelon form column rank = row rank. It should be noted here that reduced row echelon form is unique because if B1 and B2 derive from A by elementary row operations, B1<>B2 by elementary row operations and so B1=B2 if they are in reduced row echelon form becomes obvious if you try it. 

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