What is a TENSOR? (Really this time!)

The definition of a tensor made with the transformation rules of tensor components never resonated with me. The definition provided by the geometric viewpoint is much more satisfying, as you would expect since tensors are geometric objects! References: Kip Thorne, Roger Blandford: Modern Classical Physics 2017 - Chapters 1, 2 This is an amazing book that goes over all the fundamentals, by using the geometric viewpoint for physics! (and it is by Kip Thorne sooo) For anyone looking to learn GR, the best first resource: Frederic Schuller's Lecture series "Gravity and Light", Lecture 3 on Multilinear algebra:    • Lecture 3: Multilinear Algebra (Internatio...   for the full series see the playlist:    • Central Lecture Course   C. Misner, K. Thorne, J. Wheeler: Gravitation 2017 - Chapters 2, 3, 8 If you're willing to tackle this legendary 1400 page book, it will surely change your physics life. Frederic Schuller's Lecture series "The Geometrical Anatomy of Theoretical Physics", Lecture 8 on Tensor space theory over fields:    • Tensor space theory I: over a field - Lec ...   Chapters: 0:00 - 02:20 - What is a (0,2) tensor 02:20 - 03:21 - Familiar example of a tensor 03:21 - 04:27 - Multilinearity of the slots 04:27 - 05:33 - Cross product as a tensor 05:33 - 13:29 - What is a vector space 13:29 - 19:02 - Surprising examples of vectors 19:02 - 23:14 - Another example for a tensor 23:14 - 27:24 - General linear maps 27:24 - 29:05 - Dual vector spaces, covectors 29:05 - 34:25 - Familiar examples of covectors 34:25 - 36:07 - General definition of tensors 36:07 - 38:18 - Cross product as a tensor again 38:18 - 47:09 - Coordinates, components of tensors 47:09 - 52:29 - Einstein summation convention, slot naming notation 52:29 - 59:23 - Transformation of tensor components