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Let $A$ be an $m \times n$ matrix, prove that Row($A$) = Row($R$) where $R$ is the rref of $A$.

The elementary matrix theorem could be useful, so

$A = E_k ... E_1 R$, where $E_k , ... , E_1$ are elementary matrices from a single ERO, how can I proceed with elementary linear algebra?

Amad27
  • 10,465
  • If $A=BR$, where $B$ is an invertible matrix, what can you say about the row spaces of $A$ and $R$ now? Try to construct a one-one onto map from one row space to the other (of course, you require $B$ for the definition). – Sarvesh Ravichandran Iyer Apr 08 '17 at 04:56
  • How do the elementary row operations change the rows of $A$ to form the rows of $R$? Can these ever yield vectors that aren't in the row space of $A$? – Jon Warneke Apr 08 '17 at 05:00
  • See http://math.stackexchange.com/q/1998138/265466. – amd Apr 08 '17 at 05:15

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