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(b) Soit (donc tel que est un produit scalaire). Existe-t-il une valeur de telle que la base canonique de est orthogonale pour ? Et orthonormée?

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Pour que la base canonique de soit orthogonale et orthonormée pour le produit scalaire , la valeur de doit être 0.

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Bonus Knowledge

Pour que la base canonique de soit orthogonale pour le produit scalaire , il faut que pour , où sont les vecteurs de la base canonique. Cela implique que les coefficients correspondants dans la définition de doivent être nuls pour les combinaisons de ces vecteurs. En revanche, pour l’orthonormalité, il faut que .
En choisissant judicieusement , par exemple en fixant , on pourrait vérifier que les vecteurs sont orthogonaux. Cependant, pour que tout soit nécessairement orthonormé, d’autres ajustements seront nécessaires au niveau de la normalisation des vecteurs en tenant compte de la définition de .

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