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I need to prove

$$Sa \times Sb = \det(S)S^{-T}(a \times b),$$

given that $a$ and $b$ are vectors and $S$ is a second-order (rank $2$) tensor.

I have the hint: vectors $u=v$ iff $u\cdot a=v\cdot a$ for all vectors $a$.

I literally have no idea where to start with this so any help would be greatly appreciated :)

Christoph
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countduckula
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  • Did you try to use the hint? Take the dot product with a vector $v$ on both sides? It might help that $(a\times b)\cdot v = \det(a,b,v)$. – Christoph Feb 08 '19 at 16:16
  • Using (a X b) •v I get (Sa X Sb) • Sv = det(S)((a X b) • v ) how do I now get rid of the v and get S^-T – countduckula Feb 08 '19 at 16:29

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