This is so true! For machine-learning and data science, tensors are largely what is called n-dimensional array. Numpy is a popular python library built basical

This is so true! For machine-learning and data science, tensors are largely what is called n-dimensional array. Numpy is a popular python library built basically around this data type. Mathematical tensors in a particular base (sorry for the sloppy language, theoretical physicist here) can be displayed with such a data type, depsite the n-dimensional array data type (as available in numpy) lacks the (co)vector properties. Funny enough, in theoretical physics there are two groups of people: The one prefering to write out complicated objects (i.e. tensors) with indices and the ones who do not (typically called abstract tensor notation in general relativity). The latter are much more close to mathematical physics and prefer to understand tensors as mappings from (co)vector spaces to scalars, similar to as you already wrote. Most general relativity computer codes are written by the index people, not by the mathematical physicists ;-)