Is there a connection between row reduction and linear maps?

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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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