Could you help me please to move forward with the problem.
I'm trying to show that a function $\varphi_{\sigma }: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$
$\varphi_{\sigma }(x_{1}, x_{2}, ... x_{n}) = (x_{i_{1}}, x_{i_{2}}, ... x_{i_{n}})$ is a linear transformation.
Where $\sigma =\begin{bmatrix} 1 &2 & ... &n \\ i_{1} & i_{2} & ... & i_{n} \end{bmatrix}$ from $S_{n}$ is a permutation.
To show it I need to check 2 conditions: 1. that sum of vectors under transformation and sum of transformations of two vectors are equal; 2. and product with scalar of a vector under transformation is equal to the product of the transformation of the vector with scalar.
So I have chosen some other vector y and try to test first condition.
I got $\varphi_{\sigma }(x_{1} + y_{1}, x_{2} + y_{2}, ... , x_{n} + y_{n}) = ..$ and It must be equal to $(x_{i_{1}} + y_{i_{1}}, x_{i_{2}} + y_{i_{2}}, ... x_{i_{n}} + y_{i_{n}}) $ But can I just put "=" between them? I don't know how to make this step. Is it that obvious? I mean, I can say, let's say $x_{1} + y_{1} = z_{1}$ which goes to $z_{i_{1}}$ but how should I then show that it equals to the sum of transformations of x and y?