What are injective and surjective maps in vector spaces?

[ylem]

Hello! I hope I've posted this in the correct section...

I'm a 3rd year undergraduate and we're currently studying Vector Spaces (in QM) and I just don't understand what injective (one-to-one) and surjective (onto) mean? As a result I have no idea what an isomorphism is!

I realize this is probably a very simple question, but I'm just struggling so much with the course!

Cheers, Samantha

 

[arildno]

Ok, the first thing you have two focus on, is that you have two SETS, call them A and B.
Both A and B has elements, determinable by some criterion.


Now, a MAPPING from A to B takes each element in A and "relates" it to some unique element in B.

To say that a mapping is injective means that there are no two elements in A that are related to the same element in B. Thus, knowing the mapping procedure along with the element in B, we can DEDUCE from this what is the element in A which is related to the known element in A.
If we denote the element in B related to element x in A with f(x), this means that if f(x)=f(y), then x=y (only ONE unique element in A is related to the value of f(x))

To say that a map is SURJECTIVE means that whatever element Y in B you pick out, there exist an x in A so that Y=f(x).
The map covers B, so to speak.

Is this clear?

 

[ylem]

Yeah! Thanks a lot :-)