Counting Elements in Hom(V,W) for Finite Linear Transformations

In summary, Hom(V,W) is the collection of all linear transformations from an n-dimensional F-space V to an n-dimensional F-space W, where F is a finite field of prime order p. Therefore, the number of elements in Hom(V,W) is p^n. This can be proven using the fact that the dimension of Hom(V,W) is equal to the product of the dimensions of V and W. So, the answer is still n^2.
  • #1
JaysFan31

Homework Statement


The set Hom(V,W) is the collection of all linear transformations from the F-space V to the F-space W. Suppose that F,V, and W are all finite. Suppose that F=Zp for some prime p, that V is n-dimensional over F, and W is n-dimensional over F. How many elements does Hom(V,W) have?

Homework Equations


Nothing.

The Attempt at a Solution


I'm pretty sure it's p^n.

I have the proof of dimV=m and dimU=n meaning dimHom(V,U)=mn. How do I transform this proof to the one I want?
 
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  • #2
Would the answer still just be n^2?
 

1. How do you count the number of elements in Hom(V,W) for finite linear transformations?

To count the number of elements in Hom(V,W) for finite linear transformations, you need to consider the dimensions of both vector spaces V and W. Let dim(V) = m and dim(W) = n. The number of elements in Hom(V,W) is then equal to n^m, since each linear transformation from V to W can be represented by an m x n matrix.

2. What is the difference between Hom(V,W) and Hom(W,V)?

Hom(V,W) represents the set of all linear transformations from V to W, while Hom(W,V) represents the set of all linear transformations from W to V. The main difference is the direction of the linear transformations - one goes from V to W, while the other goes from W to V.

3. Can the number of elements in Hom(V,W) be infinite?

No, the number of elements in Hom(V,W) is finite for finite vector spaces V and W. This is because each linear transformation can be represented by a finite matrix, and the number of possible matrices is always finite.

4. How do you determine the dimension of Hom(V,W)?

The dimension of Hom(V,W) is equal to the product of the dimensions of V and W, i.e. dim(Hom(V,W)) = dim(V) x dim(W). This is because each linear transformation can be represented by an m x n matrix, where m is the dimension of V and n is the dimension of W.

5. What is the significance of counting elements in Hom(V,W)?

Counting elements in Hom(V,W) is important in understanding the space of all possible linear transformations between two vector spaces. It can also help in solving problems related to linear transformations, such as determining if a transformation is injective or surjective.

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