What are injective and surjective maps in vector spaces?

AI Thread Summary
Injective maps in vector spaces ensure that each element in set A corresponds to a unique element in set B, meaning no two elements in A map to the same element in B. This property allows for deducing the original element in A from its image in B. Surjective maps, on the other hand, guarantee that every element in set B is covered by some element in set A, indicating that the mapping reaches all elements in B. Understanding these concepts is crucial for grasping isomorphisms, which are mappings that are both injective and surjective. Clarity on these definitions is essential for progressing in vector space studies.
ylem
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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
 
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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?
 
Yeah! Thanks a lot :-)
 
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