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.

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)

Of course that's............... the same as a "function" from ℝⁿᵐ to ℝᵐᵖ

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