I can visualize a first-order tensor as a segment (a vector), but I'm not sure how to visualize a second-order tensor. The book that I'm trying to study is "Vector and tensor analysis with applications" (A. Borisenko and I. Tarapov)
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For a vector space $V$ over a scalar field ${\Bbb F}$, you can see it as a bilinear transformation $$B:V\times V\longrightarrow{\Bbb F}$$ which allow you to impose a metric in $V$. If we consider that $V$ is spawned by a basis $\{b_i\}$, then it is necessary that the matrix $$[B]=[B(b_i,b_j)]$$ complies certain conditions.
janmarqz
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1What conditions would these be, precisely ? – Tristan Duquesne Nov 24 '19 at 16:18
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@TristanDuquesne: that the matrix be symmetric, positive-defined and non-degenerated – janmarqz Nov 24 '19 at 16:32
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Apparently rotation and scaling matrices can be considered tensors of rank two. This is what the matrix in a affine transformation does. Look at this answer here
Zach466920
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