What is the relationship between invertible linear mappings and rank in proofs?

[Mona1990]

1. Hi!
I was wondering if anyone could help me to solve the following problem!
Let L : [R][n] ->[R][m] and M :[R][m]-> [R][m] be linear mappings.
Prove that if M is invertible, then rank (M o L) = rank (L)

thanks! :)

 

[HallsofIvy]

If M is invertible it maps R^m one-to-one onto R^m. In particular, it maps any k dimensional subset of R^m onto a k dimensional subset of R^m. Now, what does "rank" mean?

 

[Mona1990]

the dimension of the column space of M is the rank of M

and we know that dim (null M)= 0 since the null space of M is just the zero vector
and since rank = m - (dimension of the null space of M)
so rank is m?