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Why is the Column space of a matrix and its reduced row echelon form different? If for a 3X3 matrix A the first two columns are independent and the third column is dependent, then the first two columns span the column space of A and since it was obtained by reversing the row operations that led to the reduced row echelon form(R) which were linear operations, the same columns span the column space of the R.

Orpheus
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Elementary row operations are given by invertible linear transformations, and thus preserve dependency relations of the columns.

On the other hand, applying a linear transformation can change the column space itself.

  • Can you elaborate on this?...Isn't the basis for the original matrix and its row reduced echelon form same....so how is the column space different?....Or it there some sort of geometric implication that is not explicitly stated... – Orpheus Dec 13 '20 at 15:16
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    The bases for the column space and that of the rref are not in general the same. The correspondence is that if certain of the columns of the rref are a basis for the column space of it, then the corresponding columns of the original matrix are a basis for its column space. The two column spaces will be different in general, because, an invertible linear transformation can map a vector space to a different vector space, rather easily. –  Dec 13 '20 at 15:20
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    What is happening to the column space when we perform linear transformations on the rows? – Orpheus Dec 13 '20 at 15:28
  • Lots of things can happen. If the columns were eigenvectors of the linear transformations, then the column space would be preserved. But this is not necessarily the case. –  Dec 13 '20 at 15:29
  • So Eigen Vectors is the only case when the row operations performed does not change the column space...Coz I know that eigen vectors remain oriented in the same direction in the transformed space. – Orpheus Dec 13 '20 at 15:31
  • Well yes, you are right about eigenvectors not changing direction, so to speak. It is concerivable that there are other cases when the column space might come out the same. We have to be careful about generalizing. For instance, a permutation matrix should do it. –  Dec 13 '20 at 15:34
  • Thanks a lot @Chris Custer – Orpheus Dec 13 '20 at 15:38
  • Thank you, @Orpheus –  Dec 13 '20 at 15:41