What are injective and surjective maps in vector spaces?

In summary, Samantha is a 3rd year undergraduate studying Vector Spaces and is struggling to understand the concepts of injective, surjective, and isomorphism. Injective means that a mapping from set A to set B has no two elements in A that are related to the same element in B. Surjective means that for any element in set B, there is a corresponding element in set A that is related to it. This is also known as being "onto". An isomorphism is a type of mapping that preserves the structure of sets, meaning that the elements are related in a one-to-one manner.
  • #1
ylem
32
1
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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  • #2
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?
 
  • #3
Yeah! Thanks a lot :-)
 

1. What is the difference between injective and surjective functions?

Injective functions, also known as one-to-one functions, map each input to a unique output. This means that no two different inputs can produce the same output. Surjective functions, also known as onto functions, map each output to at least one input. This means that every element in the range has at least one corresponding element in the domain.

2. How can you determine if a function is injective or surjective?

To determine if a function is injective, you can use the horizontal line test. This means that if you draw a horizontal line anywhere on the graph of the function, it should only intersect the graph at most once. To determine if a function is surjective, you can check if every element in the range has at least one corresponding element in the domain.

3. Can a function be both injective and surjective?

Yes, a function can be both injective and surjective. This type of function is known as a bijective function. It means that every element in the range has exactly one corresponding element in the domain, and every element in the domain has at least one corresponding element in the range.

4. What is the significance of injective and surjective functions in mathematics and science?

Injective and surjective functions are important concepts in mathematics and science as they allow us to understand the relationship between sets and their elements. They are used in various areas such as data analysis, cryptography, and optimization problems.

5. Can a function be neither injective nor surjective?

Yes, a function can be neither injective nor surjective. This type of function does not have a one-to-one correspondence between the elements in the domain and the elements in the range. This can happen when multiple inputs map to the same output, or when not all elements in the range have a corresponding element in the domain.

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