Field of modulo p equiv classes, how injective linear map -> surjectivity

In summary, the conversation discusses the relationship between the field of modulo p equivalence classes and linear maps. It is stated that the vector space structure is irrelevant in this case. The conversation then mentions that the proof for finite-dimensional real vector spaces can be applied in this scenario, and that the fact that the field is finite makes the proof simpler. The conversation also touches on the concept of isomorphisms and mentions that if the basis for V is mapped to a basis for W, it proves that V and W are isomorphic and the linear map is an isomorphism.
  • #1
hh13
Field of modulo p equiv classes, how injective linear map --> surjectivity

Homework Statement



Let Fp be the field of modulo p equivalence classes on Z. Recall that |Fp| = p.

Let L: Fpn-->Fpn be a linear map. Prove that L is injective if and only if L is surjective.

Homework Equations



|Fpn| = pn. We also know that for an F-linear map L: V-->V, that L is onto iff {v1,...,vm} spans V, given that DimV=m<infinity and that {v1,...,vm} are column vectors of the mxm matrix associated with L.

And that L is injective iff {v1,...,vm} is linearly independent.

And that L is bijective iff {v1,...,vm} is a basis.

The Attempt at a Solution



I just don't know what it means for a linear map to have the field of modulo p equivalence classes. on Z, let alone try and do this proof. If we suppose L is injective, then kerL=0. If L is NOT surjective, then there exists some v' in Fpn such that L(v)[tex]\neq[/tex] v' for every v in Fpn. I believe this has something to do with the field of modulo p equivalence classes, but again our professor hasn't really taught us those...

Thanks =)
 
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  • #2


It's a trick question; the vector space structure is irrelevant!

Actually, IIRC, you could use the exact same proof that you would use for finite-dimensional real vector spaces. The field of scalars is irrelevant.

However, the fact you're working with a finite field makes things that much simpler.
 
  • #3


But how do we do that?

I'm having a hard time trying to go about proving linear maps are isomorphisms. I've heard that if, in a linear map L: V--W, if the basis for V is {v1,...,vn}, then if {L(v1),...,L(vn)} works as a basis for W, then it proves that V and W are isomorphic to each other and so L is an isomorphism...but I don't get why...or maybe I've heard wrong?
 

FAQ: Field of modulo p equiv classes, how injective linear map -> surjectivity

1. What is a field of modulo p equiv classes?

A field of modulo p equiv classes refers to a mathematical structure that consists of a finite set of elements, where the arithmetic operations of addition and multiplication are performed modulo a prime number p. In other words, the elements in this field are equivalence classes of integers that have the same remainder when divided by p.

2. How do injective linear maps relate to surjectivity in this context?

An injective linear map is a function that preserves linear combinations, meaning that the output of the function for a linear combination of inputs is equal to the linear combination of the outputs. In the context of a field of modulo p equiv classes, an injective linear map allows us to map the elements of one field to another. Surjectivity, on the other hand, ensures that all elements in the target field have a corresponding element in the original field. Therefore, an injective linear map guarantees that the target field is at least as large as the original field, while surjectivity ensures that the target field is the same size as the original field.

3. Can you give an example of a field of modulo p equiv classes?

Yes, an example of a field of modulo p equiv classes is the finite field of integers modulo p, denoted as GF(p). This field consists of p elements, with the arithmetic operations of addition and multiplication performed modulo p. For example, GF(5) would consist of the elements {0, 1, 2, 3, 4} with the following operations:

0 + 1 = 1, 0 * 1 = 0

1 + 3 = 4, 1 * 3 = 3

2 + 2 = 4, 2 * 2 = 4

3 + 4 = 2, 3 * 4 = 2

4 + 0 = 4, 4 * 0 = 0

4. How are fields of modulo p equiv classes used in mathematics?

Fields of modulo p equiv classes have many applications in mathematics, particularly in number theory and algebra. They are used to study properties of prime numbers, solve equations in finite fields, and study group theory. They also have applications in cryptography, as they provide a way to perform secure computations using modular arithmetic.

5. Are there any limitations to using fields of modulo p equiv classes?

One limitation of using fields of modulo p equiv classes is that they are only defined for prime numbers p. This means that the size of the field is limited, and certain operations, such as division, may not be well-defined. Additionally, some mathematical problems may be difficult to solve within this context, as the finite nature of the field may not provide enough information to solve them. However, fields of modulo p equiv classes are still widely used and have many practical applications in mathematics and computer science.

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