# Book recommendation for rigorous multilinear algebra , tensor analysis, manifolds.

I am looking for recommendation on books about multilinear algebra, tensor analysis, manifolds theory, basically everything to be able to understand basic concepts of general relativity.

I am currently following two online courses by Prof Shuller on youtube ( https://www.youtube.com/channel/UC6SaWe7xeOp31Vo8cQG1oXw/videos ). One of the course is very complete, it starts from propositional logic all the way to differential geometry. The course is really packed cause obviously there is a lot to cover.

Prof Shuller make some really good point throughout the course, and at one point he explain that taking the determinant of a matrix is not always defined (i.e, it is defined if the matrix represent an endomorphism, but not define if it represent a bilinear map see : https://youtu.be/4l-qzZOZt50?t=1h58m24s) Such remark makes me want to understand the subject in a deep manner.

I would like to possibly have a rigorous book that treat the subject coordinate free first and then explain applications using matrix and coordinates, that will cover approximatively the same content as the lectures of Prof Shuller (basically having something to dive deeper).

Any reference will be greatly appreciated !! Note : I already contacted Prof shuller with no luck.

Thanks !

• A classic (and cheap!) reference is Bishop and Goldberg's Tensor Analysis on Manifolds. – symplectomorphic Jan 8 '16 at 10:38
• The book looks really promising by the look of it ! I looked at the determinant chapter and well it is not defined in terms of matrices (which is a nice point for me) Thank you ! – user149705 Jan 8 '16 at 15:26
• you ought to consider nowadays that the tensor product is a construction that converts the category of vector spaces into a monoidal category – janmarqz Jan 9 '16 at 3:53
• but also take a look at mathoverflow.net/questions/17521/… – janmarqz Jan 9 '16 at 4:00