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.

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So, 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.

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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

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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

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Of course that's............... the same as a "function" from ℝⁿᡐ to ℝᡐᡖ

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