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Finishing 7/3/23 Allgèbre linéaire

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Todo: Reorganise chapters
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Debucquoy Anthony
2023-04-20 23:26:55 +02:00
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@ -13,11 +13,19 @@ Nous avons vu les fonctions:
## Application Linéaire
- Soient \\( V_1, V_2 \subseteq \mathbb{R}^n \\)
- On dit que \\( L: V_1 \to V_2 est une **Application Linéaire** \\) Ssi
1) \\( \forall u, v \in V_1 \quad L(U+v) = L(u) + L(v) \\)
- On dit que \\( L: V_1 \to V_2\\) est une **Application Linéaire** ssi
1) \\( \forall u, v \in V_1 \quad L(u+v) = L(u) + L(v) \\)
2) \\( \forall \lambda \in \mathbb{R} \quad L(\lambda v) = \lambda L(v) \\)
| Exemples | Contre-Exemples |
| --------- | --------------- |
| L(x) = x | L(x) = \|x\| |
| L(x) = 2x | L(x) = 2x + 1 |
| | L(x) = x² |
| | L(x) = sin(x) |
## Image et Noyaux
- Soit \\( L: V_1 \to V_2 \\) Une application Linéaire
- \\( Ker(L) = \\{ v \in V_1 \mid L(v)=0 \\} \\) Noyau de L
- \\( Im(L) = \\{ v \in V_2 \mid \exists u \in V_1 \quad L(u) = v \\} \\) Image de L
- \\( Im(L) = \\{ v \in V_2 \mid \exists u \in V_1 \quad L(u) = v \\} \\) Image de L