Is there a connection between row reduction and linear maps?

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SUMMARY

The discussion centers on the relationship between a matrix \( M \) viewed as a linear map \( L \) and its row-reduced form \( M' \) viewed as a linear map \( L' \). It is established that while both maps span the same space, they are not identical. Specifically, any invertible matrix can be row reduced to the identity matrix, confirming that both forms share the same image, which is the row space.

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WWGD
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Hi, everyone:

I was wondering about the relation of a matrix , seen as a linear map,
and its row-reduced form, seen the same way.

More specifically: take a matrix M , seen as a linear map L. What
is the relation between this map L given by M, and the linear map L'
given by M', the row-reduced form of M. Are these maps L,L' equal?
Are L,L' related in some other way?.

Thanks.
 
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Indeed, any invertible matrix can be row reduced to the identity matrix, precisely because they both have the entire space as image (row space).
 

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