Let $n \geqslant 2$. Define the map τ from $\mathbb{K}^n$ × ... × $\mathbb{K}^n$ (where the factor $\mathbb{K}^n$ appears $n$ times) to $\mathbb{K}$ by $$τ ((x_{1},i)^n _{i=1}, ...,(x_{n},i)^n _{i=1} ) := x_{1,1} + x_{2,2} + ... + x_{n,n}.$$
Is the map τ : $\mathbb{K}^n$ × ... × $\mathbb{K}^n$ → $\mathbb{K}$ multilinear?
I have been provided with this solution:
No, τ is not multilinear. For example, if we let $(x_{1,1} + x_{2,2} + ... + x_{n,n}) ∈ \mathbb{K}^n$ have all zero entries except $x_{1,1} = 1$, fix all columns except the second one $x′_{2}$ is allowed to vary, then the function τ as a function of $x′_{2}$ is $x′_{2}$→ 1 + $x′_{2,2}$ . This function is not a linear map $\mathbb{K}^n$ → $\mathbb{K}$ (even though it is affine) so the map τ is not multilinear.
Is there a more systematic approach to the answer?