Suppose $F$ is a field with $p^n$ elements where $p$ is prime. Let $V$ be a $k$-dimensional vector space over $F$.

finite-fieldslinear algebralinear-transformationsvector-spaces

Suppose $F$ is a finite field with $p^n$ elements where $p$ is prime. Let $V$ be a $k$-dimensional vector space over $F$. Then what is the cardinality of the following?

  1. The number of linear transformations $T:V \rightarrow V$
  2. The number of invertible linear transformations $T:V \rightarrow V$
  3. The number of linear transformations $T:V \rightarrow V$ with determinant 1.

I have tried to solve it by trying to count the number of matrix representations (of linear transformations), but even if I manage to do that, a linear transformation can have more than one matrix representation.

Best Answer

Hint

A linear map is fully and uniquely defined by the image of the canonical basis $\{e_1, \dots, e_k\}$. And there are $(p^n)^k = p^{nk}$ vectors in $V$.

Point 1

Each $e_i$ may have $ p^{nk}$ images. Therefore there are $p^{nk^2}$ linear transformations.

Point 2

The image of $e_1$ can be every vector except the zero one. The image of $e_2$ can be what you want providing that it doesn't lie in the image of $e_1$. Similarly for the other vectors.

Point 3

The determinant is a group homomorphism from $\mathcal L(V,V)$ to $F$. Its kernel if $\{1\}$.

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