How to show a mapping T is linear. I must show that $T :\mathbb{R}^2→\mathbb{R}^2$ given by $T(x, y)=(x + y, x)$;
I know that a mapping is linear if:

(1) $T(x+y) = T(x)+T(y)$ 

and

(2) $T(\lambda x) = \lambda T(x)$ 

or 

(3) $T(\lambda x + \rho y) = \lambda T(x) + \rho T(y)$

I don't know how to implement (3).

$T(\lambda(x,y))=T(\lambda x,\lambda y)=(\lambda x+\lambda y,\lambda x)=\lambda(x+y,x)=\lambda T(x,y)$ 
and also
$T((a,b)+(x,y))=T((a+x,b+y))=(a+b+x+y,a+x)=(a+b,a)+(x+y,x)=T((a,b))+T((x,y))$
So (1) and (2) are satisfied and this concludes that the mapping is linear.
