What is the easiest way to understand tensors for someone who understands linear algebra?

Answer 1

Start with this:

1. The difference between an ordered pair of vectors and a tensor product of two vectors is that if you multiply one of the two vectors by a nonzero scalar and the other by the reciprocal of that scalar, you get a different ordered pair but you still have the same tensor product.
2. A tensor is a sum of such tensor products.