Linear Mappings

Let$(E, +_E, \cdot_E)$ and $(F, +_F, \cdot_F)$ be two vector spaces over a commutative field $K$. \textbf{Definition 2.1.1.} A mapping $f: E \to F$ is called a linear mapping from $E$ to $F$ if it satisfies: \begin{enumerate} \item For all $x, y \in E$, $f(x +_E y) = f(x) +_F f$; \item For all $a \in K$ and $x \in E$, $f(a \cdot_E x) = a \cdot_F f(x)$. \end{enumerate} The set of all linear mappings from $E$ to $F$ is denoted by $L(E, F)$.

Remark: Let \( E \) and \( F \) be two \( K \)-vector spaces, and let \( f \) be a linear map. Let \( o \) be the zero vector of \( E \).\( Remark: Let \( E \) and \( F \) be two \( K \)-vector spaces, and let \( f \) be a linear map. Let  \(0 _{E} \) be the zero vector of \( E \). \),\(0 _{E} \) be the zero vector of \( E \). \)f(0 _{E})=0 _{F}.

1. \(f(0 _{E})=(0 _{F}\)
2. For x\in E, We denote by −X the symmetric element of X with respect to the additive operation  + in E.

Notation: Let \( E \) and \( F \) be two \( K \)-vector spaces,The set of all linear mappings from \( E \) to \( F \) is denoted by \( \mathcal{L}_{K}(E, F) \) instead of \( \mathcal{L}(E, F) \).

Moreover, if \( E \) and \( F \) are finite-dimensional, then \( \mathcal{L}(E, F) \) is finite-dimensional and the dimension of \( \mathcal{L}(E, F) \) =

Particular case: Let E and F be two K-vector spaces, and let $f$

a.If \( E = F \), we say that \( f \) is an endomorphism of the vector space \( E \). The space of endomorphisms is denoted by

$b.$$If$\( f \) is bijective, we say that \( f \) is an isomorphism from \( E \) to \( F \). We say that the two \( K \)-vector spaces \( E \) and \( F \) are isomorphic, and we write

$c.$$If$\( E = F \) and \( f \) is bijective, we say that \( f \) is an automorphism of \( E \).