Dimension & Linear Maps: Does U=V?

[FunkyDwarf]

Hey guys,

Does dimension remain unchanged under a linear map? Ie if i have a map f:U->V does dim(U) = dim(img(f))?

Cheers
-Z

 

[genneth]

Not necessarily.

 

[morphism]

For a trivial example, take the zero map on a space of positive dimension.

Something that might be of interest to you is the rank-nullity theorem.

 

[FunkyDwarf]

Yeh i kinda figured it didnt hold. Actually i was going to use it to prove the RL theorem if it was true

cheers
Z

 

[HallsofIvy]

If L is linear, L(U) cannot have dimension higher than U. It can have dimension lower than U. Of course, to do that, L must map many vectors to the 0 vector: its kernel is not empty. The "nullity-rank" theorem morphism mentioned says that the dimension of L(U) plus the dimension of the kernel of U is equal to the dimension of U.