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 (Inter...  
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 -...  


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