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If V is an inner product space, then it is possible to define the outer product as a linear map V → W. In this case, the linear map x ↦ v , x {\displaystyle \mathbf {x} \mapsto \langle \mathbf {v} ,\mathbf {x} \rangle } is an element of the dual space of V , as this maps linearly a vector into its underlying field, of which v , x ...
In the special case v i = w i, the inner product is the square norm of the k-vector, given by the determinant of the Gramian matrix ( v i, v j ). This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on ().
On an inner product space, or more generally a vector space with a nondegenerate form (hence an isomorphism ), vectors can be sent to covectors (in coordinates, via transpose), so that one can take the inner product and outer product of two vectors—not simply of a vector and a covector.
The regressive product, like the exterior product, is associative. [28] The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper gives a full treatment of several different inner products developed for geometric algebras and their interrelationships, and the notation is taken from there. Many ...
It is often called the inner product (or rarely the projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
For example, the inner product of a polar vector and an axial vector resulting from the cross product in the triple product should result in a pseudoscalar, a result which is more obvious if the calculation is framed as the exterior product of a vector and bivector. They generalise to other dimensions; in particular bivectors can be used to ...
Topological tensor product. The tensor product, outer product and Kronecker product all convey the same general idea. The differences between these are that the Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its intrinsic definition. The outer ...
an outer semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products.