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we introduced multilinear maps today with the following definitions:

Let $V$ be a finite-dimensional real vector-space and $k,l\in \mathbb{N}$. A map $T:V\times...\times V\cong V^k\rightarrow \mathbb{R}$ is multilinear, if it is linear in every argument, which means $\forall v_1,...,v_k,w\in V, \lambda,\mu\in \mathbb{R}$ and $i\in \{1,...,k\}$ the following is true: $$T(v_1,...,\lambda v_i+\mu w,...,v_k)=\lambda T(v_1,...,v_i,...,v_k)+\mu T(v_1,...,w,....,v_k)$$ The space of multilinear maps $T:V^k\rightarrow \mathbb{R}$ is $T^k(V)$. For $S\in T^k(V)$ and $T\in T^l(V)$ we define $(S\otimes T)\in T^{k+l}(V)$ with:

$$(S\otimes T)(v_1,...,v_k,v_{k+1},...,v_{k+l})=S(v_1,...,v_k)T(v_{k+1},...,v_{k+l})$$ We also wrote down:

In general $S\otimes T\neq T\otimes S$.

I tried to find an example for that but I couldn't. Can someone give me an example? Thanks in advance.

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Choose k and l to be 1 and $V = \mathbb{R}^2$. Let S map $e_1$ to zero and $e_2$ to 1 and T do the opposite. Evaluate the tensor products at $(e_1, e_2)$.

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