Hello.

Definition: A is convex if and only if $\displaystyle \forall{x_1,x_2\in{A}}$ the segment which unite them $\displaystyle (\lambda x_1 + (1-\lambda)x_2 ) \in{A}; 0\leq{\lambda\leq{1}}$.

So, let consider the set of vectors $\displaystyle By$ where B is a n x m matrix and $\displaystyle y = (y_1,y_2,...,y_m): y_i\geq{0}$ is an m-vector : $\displaystyle y_1+y_2+...+y_m=1$, prove that By is a convex set.

Thanks a lot.