Joining a discussion on Linear Algebra

Happy new year, Vlado!  Thank you so much for posting Chapter I of your book Elastoplasticity Theory.  I really like your book.  It's modern and comprehensive, covering many difficult aspects of our field.  It is also very pedagogical.   

In the chapter you posted here, you have developed tensors using dyads.  Several other approaches have been mentioned in the two recent threads:

• Textbook on linear algebra
• Scalars done wrong

Here is a list of the approaches mentioned:

1. Use dyads.
2. Specify rules to transform components associated with a change of basis.
3. Define a tensor as a multilinear linear function on a vector space.
4. Define a tensor as a multilinear map from a collection of vector spaces to yet another vector space.

I believe that approaches 1-3 are equivalent.  Approach 4 is a natural generalization of 3, but is perhaps too general for common applications in mechanics.

Now saying that approaches 1-3 are equivalent overlooks one significant aspect:  they look different.  We need to teach and students need to learn.  Which approach is good for learning a subject for the first time or second time?  Should we learn them all?

The experience of teachers and students is important. It will be useful if people can talk about their experience.