What is the most common and appropriate definition of tensor?

Question

Tensor can be thought of as generalization vectors but tensor is described in many ways sometimes an array of numbers.

What is the most common and appropriate definition of tensor ?

Answer 1

I favor the abstract definition of a tensor space as a quotient vector space.

We define the tensor product of vector speces $V$ and $W$ over a common base field as the quotient vector space: $$ V \otimes W := F(V\times W)/\sim$$ where $F(Z)$ is a free vector space generated by elements of set $Z$, and $\sim$ is the minimimal equivalence relation such that

• $(v,w)+(v',w) \sim (v+v',w)$ and $(v,w)+(v,w') \sim (v,w+w')$
• $(\lambda v, w) \sim \lambda(v,w) \sim (v,\lambda w)$

This definition can be generalized to define a tensor product of arbitrary number of vector spaces. A tensor is an element of a tensor space. We define $v\otimes w := [(v,w)]_\sim$

I like this definition because from it immediately follow the important arithmetic properties of tensors:

• $ (v\otimes w)+(v'\otimes w) = (v+v')\otimes w $ and $ (v\otimes w)+(v\otimes w') = v\otimes (w+w') $
• $ (\lambda v)\otimes w = \lambda(v\otimes w) = v \otimes (\lambda w)$

It also shows that a tensor product is uniquely defined and independent of the bases of the vector spaces. But given specific bases of $V$ and $W$, we can easily construct an isomorphism between the abstract tensor product and the array of numbers. We just need to remember that this isomorphism is basis-dependent.