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The thing with tensors is that they only make sense as a term in the context where they're used (diffgeo), and if you try to explain them in terms of vector analysis, the point gets quickly lost.

In the context of vector analysis, \(V=\mathbb{R}^m\), so you indeed get \(V^n\) are \(m\times n\) matrices and \(V^p\) are \(m\times p\) matrices

But why does it lose all meaning in the end?

Because you can make a vector out of a matrix, using "array notation". For example,

[a b]

[c d]

is the same as

(a, b; c, d)

which in turn is the same as

(a,b,c,d)

Hannah π@ZevenKorian@mathstodon.xyzSo, get ready for the definitions!

Let \(V\) be a vector field. A tensor is basically a linear map \(T: V^n \longrightarrow V^p\) This means that \(T\) transforms matrices into matrices.