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Consider a matrix $A$. Is $ \mathrm{image}(A) $ equal to $ \mathrm{image}(\mathrm{rref}(A)) $? Their kernels and solution of the system $ Ax = b $ is the same.

Kevin Chen
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Mohamed Osama
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  • The image is the span of the columns, i believe. Row operations preserve the dimension of the image but the resulting rows do not necessarily span the image. – recmath Sep 11 '14 at 15:58
  • Yes. It's easy to see this by showing that (separately) each of the three basic row operations preserves the image, since reducing to RREF can be achieved by stringing together those operations appropriately. – Travis Willse Sep 11 '14 at 16:08

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No. Here is an example: Let A be $\left( \begin{array}{cc} 1 & 1\\ 1 & 1\\ \end{array} \right)$.

Then rref(A)=$\left( \begin{array}{cc} 1 & 1\\ 0 & 0\\ \end{array} \right)$.

Column space of A is $span\left\{\left( \begin{array}{c} 1\\ 1\\ \end{array} \right)\right\}$.

Column space of rref(A) is $span\left\{\left( \begin{array}{c} 1\\ 0\\ \end{array} \right)\right\}$.

You are right in saying that they have the same null space (kernel).

Novice
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