Linear application

In Mathematical, a linear application (also called linear operator or linear transformation ) is an application between two vector spaces which respects the addition of the vectors and the scalar multiplication defined in these vector spaces, or, in other terms, which “preserves the linear combinations”.

Definitions

That is to say

ƒ: E → F

an application where E and F is two $\backslash \; mathbb\; K$ vector spaces.

ƒ is a linear application (or morphism of $\backslash \; mathbb\; K$ vector spaces ) if:

* $\backslash \; forall\; X\; \backslash \; in\; E,\; \backslash \; forall\; there\; \backslash \; in\; E,\; F\; (x+y)\; =\; F\; (X)\; +\; F\; (there)$

* $\backslash \; forall\; \backslash \; lambda\; \backslash \; in\; \backslash \; mathbb\; K,\; \backslash \; forall\; X\; \backslash \; in\; E,\; F\; (\backslash \; lambda\; \backslash \; cdot\; X)\; =\; \backslash \; lambda\; \backslash \; cdot\; F\; (X)$

An application having the first property is known as additive, and, for the second, homogeneous.

ƒ is a Isomorphisme if:

* ƒ is a Morphisme

* ƒ is bijective

ƒ is a Endomorphisme if:

* ƒ is a morphism

* F = E

ƒ is a Automorphisme if:

* ƒ is a endomorphism

* ƒ is bijective

If $F\; =\; \backslash \; mathbb\; K$, one speaks about linear Forme .

One notes

• $L\_\; \{\backslash \; mathbb\; K\}\; (E,\; F)$ the whole of the linear applications of E in F ;
• $Isom\_\; \{\backslash \; mathbb\; K\}\; (E,\; F)$ the whole of isomorphisms of E in F ;
• $L\_\; \{\backslash \; mathbb\; K\}\; (E)$ the whole of the endomorphisms of E ;
• $GL\_\; \{\backslash \; mathbb\; K\}\; (E)$ (also called the linear group ) the whole of the automorphisms of E .

As its name indicates it, the linear group, provided with the composition, is a group.

Core and Image

If ƒ is a linear application of E in F , one defines the core of ƒ, noted Ker (ƒ) ( kern means “German core”), and the image of ƒ, noted Im (ƒ), by

$\backslash \; operatorname\; \{Ker\}\; (F)\; =\; \backslash \; \{\backslash ,\; X\; \backslash \; in\; E:\; F\; (X)\; =0\; \backslash ,\; \backslash \}$

$\backslash \; operatorname\; \{Im\}\; (F)\; =\; \backslash \; \{\backslash ,\; F\; (X):\; X\; \backslash \; in\; E\; \backslash ,\; \backslash \}$

ker (ƒ) is a vectorial subspace of E and im (ƒ) are a subspace of F .

The formula following, valid for a space E of Dimension finished, is often useful:

$$

\ dim (\ operatorname {Ker} (F)) + \ dim (\ operatorname {Im} (F))

\ dim (E) \, .

It is also called Théorème of the row .

The number dim (Im (ƒ)) is also called row of ƒ and is noted rg (ƒ). If E and F is of finished size and ƒ is represented by the matrix has , then the row of ƒ is equal to the row of the matrix has ; such a linear application is a Tenseur of order 2, once covariant, once contravariant.

Examples

• the linear function “usual”: $f:\; X\; \backslash \; mapsto\; has\; \backslash \; cdot\; x$ where has is a Scalaire;

• the linear combinations of vectors
• the derivation application:
• : D: $D\; (\backslash \; mathbb\; \{R\},\; \backslash \; mathbb\; \{R\})\; \backslash \; to\; F\; (\backslash \; mathbb\; \{R\},\; \backslash \; mathbb\; \{R\})$
• :: $f\backslash quad\; \backslash mapsto\backslash quad\; f\text{'}$

Theorems

• Theorem of Banach-Steinhaus

• Theorem of Hahn-Banach

Bonds

• vector Space normalized

• Linear algebra
• linear Equation
• System of linear equations

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