Linear Mappings

Let \( (E, +_E, \cdot_E) \) and \( (F, +_F, \cdot_F) \) be two vector spaces over a commutative field \( K \).

Definition

A mapping \( f : E \to F \) is called a linear mapping from \( E \) to \( F \) if it satisfies:

For all \( x, y \in E \),

\[f(x +_E y) = f(x) +_F f(y).\]

For all \( a \in K \) and \( x \in E \),

\[f(a \cdot_E x) = a \cdot_F f(x).\]

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 \( E \).