I'm reading the text fundamentals of linear algebra and optimization [the last paragraph in p.77], it's said the direct sum is a linear map, but I cannot see why :
Given $X$, $Y$ subspace of a vector space $E$, we define a map $$ f \left\{ \begin{array} \\ X \times Y \rightarrow E \\ (x,y) \rightarrow x+y\end{array}\right.$$ Then for any $\lambda \neq 0$ $$ f(x_1+\lambda x_2,y) = x_1+\lambda x_2+y$$ And $$ f(x_1,y)+\lambda f(x_2,y) = x_1+\lambda x_2+ (1+\lambda)y$$ So it cannot be linear on the first position (coordinate) since $f(x_1+\lambda x_2,y) \neq f(x_1,y)+\lambda f(x_2,y)$. Where am I wrong? How the text say $f$ is clearly linear?