  We give new characterizations of the algebra
$\mathscr{L}_n(\mathbb{F}_{q^n})$ formed by all linearized polynomials over the
finite field $\mathbb{F}_{q^n}$ after briefly surveying some known ones. One
isomorphism we construct is between $\mathscr{L}_n(\mathbb{F}_{q^n})$ and the
composition algebra
$\mathbb{F}_{q^n}^\vee\otimes_{\mathbb{F}_{q}}\mathbb{F}_{q^n}$. The other
isomorphism we construct is between $\mathscr{L}_n(\mathbb{F}_{q^n})$ and the
so-called Dickson matrix algebra $\mathscr{D}_n(\mathbb{F}_{q^n})$. We also
further study the relations between a linearized polynomial and its associated
Dickson matrix, generalizing a well-known criterion of Dickson on linearized
permutation polynomials. Adjugate polynomial of a linearized polynomial is then
introduced, and connections between them are discussed. Both of the new
characterizations can bring us more simple approaches to establish a special
form of representations of linearized polynomials proposed recently by several
authors. Structure of the subalgebra $\mathscr{L}_n(\mathbb{F}_{q^m})$ which
are formed by all linearized polynomials over a subfield $\mathbb{F}_{q^m}$ of
$\mathbb{F}_{q^n}$ where $m|n$ are also described.
