I have this confusion over Exterior Algebra.

Sometimes I see it defined as $\Lambda(V)$. (e.g. https://en.wikipedia.org/wiki/Exterior_algebra)

While sometimes it is defined as $\Lambda(V^*)$, where $V^*$ is the dual of the vector space $V$. (e.g. John Lee's book).

Can someone enlighten me why is this so?



One can speak of the exterior algebra of any vector space $V$, so also of the exterior algebra of its dual $V^{\ast}$. Elements of the exterior algebra of the dual vector space $V^{\ast}$ can be thought of as forms on $V$, while elements of the exterior algebra of $V$ are multivectors in $V$. (If $V$ is finite-dimensional, the $k$th exterior power $\wedge^{k}(V^{\ast})$ of its dual is naturally identified with the dual of its $k$th exterior power $\wedge^{k}(V)$.) Differential forms (presumably the context for the appearance of the exterior algebra in Jack Lee's book) are sections of the bundle whose fibers are the exterior algebras of the cotangent spaces, so in the context of differential topology it is often natural to work with $\wedge(T^{\ast}M)$.


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