Let $\mathbb{F}_{q}$ and $\mathbb{F}_{q^n}$ be two finite fields where $q$ is a prime power and $n$ is a positive integer. A linearized polynomial ($q$-polynomial) $L(x)$ is a univariate polynomial over $\mathbb{F}_{q^n}$ having the following form $$L(X)=a_0 x + a_1 x^q + \cdots + a_{n-1} x^{q^{n-1}}.$$
It is well known that $L(x)$ can be treated as a linear transformation of the vector space $\mathbb{F}_{q^n}$ which is viewed as a $n$-dimensional vector space over $\mathbb{F}_{q}$.
My task is try to demtermine the subspace $V=\{L(t)\mid t \in \mathbb{F}_{q^n}\}$ by these coefficients $a_0,a_1,\ldots,a_{n-1}$.
To make it easy understood, let $n=3$ and $L(x)=ax^{q^2}+bx^q+cx$ where $a,b,c\in \mathbb{F}_{q^n}$. How to represent the set $$V=\{ax^{q^2}+bx^q+cx \mid x \in \mathbb{F}_{q^n}\}$$ in terms of $a$, $b$ and $c$, i.e., $V=\{x\in \mathbb{F}_{q^n} \mid a,b,c \text{ meets things.}\}$?