Linear Transformation questions about dimensions

[potmobius]

1. Say you have a linear transform from A to B, and where A has a higher dimension than B. How do you show that the kernel of the transform has more than one element (i.e. 0)? Also, if B has a higher dimension than A, then how to show that the transform isn't surjective?

2. The attempt at a solution

By showing that the kernel has more than the element 0, I want to show that the transform isn't injective. But I'm not quite sure how to get there just by using the fact that A has a higher dimension than B. Is that a good way(as in, not too complicated) of proving it? Any ideas?

For the other part, it makes sense intuitively, since the basis of A will have less elements than the basis of B, so there shouldn't be a surjection. But how do you proceed from there to show that the image of A is a proper subset of B?

 

[sunjin09]

Think of the theorem that says the dim(A)=dim(Ker(f)) + dim(Im(f)), where f:A->B is a linear morphism (transformation)

 

[potmobius]

Thanks! Now I feel really stupid for not considering Rank-Nullity before asking this...

 

[sunjin09]

Thanks! Now I feel really stupid for not considering Rank-Nullity before asking this...

You're welcome. I feel stupid for not knowing that theorem has a name ...

 

[potmobius]

Do you know any bad math jokes related to this?