How does row reduction relate to a linear map's matrix?

[WWGD]

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.

 

[ice109]

they span the same space but they are not the same map

http://en.wikipedia.org/wiki/Row_equivalence#Row_space

 

[HallsofIvy]

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