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carte mentale sur les espaces vectoriels
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L'essentiel à connaitre sur les espaces et sous espaces vectoriels
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Définition :Définition :
u+v=v+u
u + (v + w) = (u + v) +w Soit E un espace
u+e=u vectoriel. F, une partie
u + u' = e de E, est un sous-espace
1xu=u vectoriel si :
a (b x u) = (a x b) u e F
u+v F Soit AX=0 un système d'équations linéaires
a (u + v) = au + av
a.u F homogènes à p variables. Alors l'ensemble des p
(a + b) u = au + bu
vecteurs solutions est un sous-espace vectoriel de R
L'intersection de deux sous-espaces vectoriels est un
sous-espace vectoriel.
Sous-espaces Mais la réunion de deux sous-espaces vectoriels
Espaces vectoriels n'est en général pas un sous-espace vectoriel.
vectoriels
F et G sont en somme directe dans E si :
F G= e
F+G=E
Somme de deux sous-espaces vectoriels : l'ensemble
des éléments u + v où u F et v G
F + G est un sous-espace vectoriel.
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