Vanille
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Hello all!
I've just started to study general relativity and I'm a bit confused about dual basis vectors.
If we have a vector space \textbf{V} and a basis \{\textbf{e}_i\}, I can define a dual basis \{\omega^i\} in \textbf{V}^* such that: \omega^i(\textbf{e}_j) = \delta^i_jBut in some pdf and documents I found this relationship: \omega^i\cdot\textbf{e}_j = \delta^i_j So I don't understand why these two relationships are equals.
In fact I know that there's an isomophism \Phi: \textbf{V}\rightarrow\textbf{V}^* induced by the inner product in such a way that: \tilde v(\textbf{u}) = \textbf{v}\cdot\textbf{u}\qquad\forall\textbf{u}\in\textbf{V}Where \tilde v is the covector associated to the vector \textbf{v} by the isomorphism \Phi.
So I expect that the basis associated to the dual basis is exactly the reciprocal basis: \omega^i(\textbf{e}_j) = \textbf{e}^i\cdot\textbf{e}_j =\delta^i_j.So the dual basis \{\omega^i\} seems to be equal to the reciprocal basis \{\textbf{e}^i\}.
I think I'm doing a very bad mistake.
Can anyone help me, please? Thank you!
I've just started to study general relativity and I'm a bit confused about dual basis vectors.
If we have a vector space \textbf{V} and a basis \{\textbf{e}_i\}, I can define a dual basis \{\omega^i\} in \textbf{V}^* such that: \omega^i(\textbf{e}_j) = \delta^i_jBut in some pdf and documents I found this relationship: \omega^i\cdot\textbf{e}_j = \delta^i_j So I don't understand why these two relationships are equals.
In fact I know that there's an isomophism \Phi: \textbf{V}\rightarrow\textbf{V}^* induced by the inner product in such a way that: \tilde v(\textbf{u}) = \textbf{v}\cdot\textbf{u}\qquad\forall\textbf{u}\in\textbf{V}Where \tilde v is the covector associated to the vector \textbf{v} by the isomorphism \Phi.
So I expect that the basis associated to the dual basis is exactly the reciprocal basis: \omega^i(\textbf{e}_j) = \textbf{e}^i\cdot\textbf{e}_j =\delta^i_j.So the dual basis \{\omega^i\} seems to be equal to the reciprocal basis \{\textbf{e}^i\}.
I think I'm doing a very bad mistake.
Can anyone help me, please? Thank you!