What's the difference between tensors and vectors?

[johann1301]

Whats the difference between tensors and vectors?

 

[slider142]

Whats the difference between tensors and vectors?

A tensor is a multilinear map between Cartesian products of vector spaces and real numbers. In more colloquial terms, a tensor assigns a real number to a list of vectors, where each vector's map is linear. An example is a tensor whose input is a list of two Euclidean vectors, and whose output is the signed area of the unique parallelogram spanned by those vectors. Another example with two vector inputs and one real output, linear in both arguments, is the dot product.
For any vector space V, there is a dual space V* consisting of all linear maps from V into the set of real numbers R. For example, in the vector space R³, the map that takes any vector of the form (x, y, z) to ax + by + cz is an element of the dual space. Notice it could be considered a dot product (a, b, c) . (x, y, z) between two vectors. This motivates the idea that the dual space and the vector space are both equally potent vector spaces, so we may give the dual space a vector space structure as well. (Of course, we prove that this can be done for any vector space, not just R³). Notice that now, each particular vector (x, y, z) in V can equally be considered a map from an arbitrary element (a, b, c) of the dual space into R.
Thus, since a single vector in a vector space V can be considered as a map from the dual space V*, which is a vector space, to R, a vector can be considered a tensor. Likewise, since a Cartesian product of vector spaces can itself be given a vector space structure, a tensor can be considered a vector. It all depends the particular context you want to work with at any particular time. So they are very similar, and in many cases can be conflated.

 

[johann1301]

Thanks for taking the time to write this!