Multilinear Algebra
نویسنده
چکیده
This project consists of a rambling introduction to some basic notions in multilinear algebra. The central purpose will be to show that the div, grad, and curl operators from vector calculus can all be thought of as special cases of a single operator on differential forms. Not all of the material presented here is required to achieve that goal. However, all of these ideas are likely to be useful for the geometry core course. Note: I will use angle brackets to denote the pairing between a vector space and its dual. That is, if V is a vector space and V ∗ is its dual, for any v ∈ V , φ ∈ V ∗, instead of writing φ(v) for the result of applying the function φ to v, I will write 〈φ, v〉. The motivation for this notation is that, given an inner product 〈·, ·〉 on V , every element φ ∈ V ∗ can be written in the form v 7→ 〈v0, ·〉 for some vector vo ∈ V . The dual space V ∗ can therefore be identified with V (via φ 7→ v0). The reason we use the term, pairing, is that duality of vector spaces is symmetric. By definition, φ is a linear function on V , but v is also a linear function on V ∗. So instead of thinking of applying φ to v, we can just as well consider v to be acting on φ. The result is the same in both cases. For this reason, it makes more sense to think of a pairing between V and V ∗: If you pair an element from a vector space with an element from its dual, you get a real number. This notation will also be used for multilinear maps.
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تاریخ انتشار 2008