Given a vector space V over a field F, a tensor is a multilinear map T: V x V x V x ... x V^* x V^* x V^* -> F You're mixing up geometric objects (multilin

Given a vector space V over a field F, a tensor is a multilinear map T: V x V x V x ... x V^* x V^* x V^* -> F You're mixing up geometric objects (multilinear transformations) with their coordinate representations. A geometric object, such as a multilinear map, is automatically invariant to the coordinate system. An array of numbers, like a matrix, is not.