Calculating Dim(ran(T) & Dim(Ker(T)

[squenshl]

I have a problem.
Calculate Dim(Ran(T)) if T is 1-to-1. Also calculate Dim(Ker(T)) if T is onto.
How do you think I should do this?

 

[matt grime]

There is nothing to calculate. You just need to think about it. A 1-1 map is an isomorphism onto its image for example - plus there are the standard results like the rank nullity theorem to help you.

 

[HallsofIvy]

If T is a linear transformation from a vector space of dimension n to a vector space of dimension m, then dim(Ran(T))+ dim(Kernel(T))= n. That's the "rank-nullity" theorem matt grime mentioned. If T is "one-to-one", then it maps only the 0 vector to the 0 vector so dim(Kernel(T))= ? If T is "onto" what is dim(Ran(T)).

dim(Ran(T)) is also called the "rank" of T and dim(Kernel(T)) is the "nullity" of T.