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I learned that a tensor is a linear map from some number of vectors to the real numbers.

For example, $t(\vec{u},\vec{v})=\vec{u}\cdot\vec{v}$ is a rank-2 tensor as it takes 2 vectors as arguments and combines them into a real number.

I then read that a rank-1 tensor is a vector. How is that possible? If the rank-1 tensor is a vector, it will mean $t({\vec u})=\vec{u}$, but $\vec{u}$ is not a real number.

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  • $\begingroup$ "If the rank-1 tensor is a vector, it will mean $t({\vec u})=\vec{u}$" How so? $\endgroup$ – Acccumulation Sep 20 at 19:54
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An $(r,s)$-tensor maps $r$ dual vectors and $s$ vectors to some field (e.g. $\mathbb{R}$). The rank of a tensor is defined as $r+s$. One can interpret a vector $v$ as $(1,0)$-tensor, mapping a dual vector $w$ to the number $w(v)$.

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