Help for proving rank of matrix

Question

This is a problem from our practice exam. Could anyone tell me how to approach this question and prove details.

Super appreciate!

Answer 1

a) An injective linear map always will preserve the dimension of the domain space, because it maps linearly independent set to linearly independent set. Therefore dim(im(T))=dim(domain space)=n where $m>n$.

b)Rank and Nullity theorem is a classical theorem in Linear algebra. dim(ker)+dim(im)=dim(domain space). T is onto, so dim(im)=m. From this you can deduce that dim(ker)=n-m where $n>m$

c) From $(a)$ and $(b)$ you can answer yourself the third question