\alpha x+\beta y=\left( \alpha x_1 + \beta y_1,...,\alpha x_n + \beta y_n\right)\\
\alpha_1\left( \alpha x_1 + \beta y_1\right) +\alpha_2\left( \alpha x_2 + \beta y_2\right)+...+\alpha_n\left( \alpha x_n + \beta y_n\right) =\\
\alpha\left( \alpha_1 x_1+...+\alpha_n x_n\right) + \beta\left( \alpha_1 y_1+...+\alpha_n y_n\right)=\alpha B + \beta B