Prove that By is a convex set

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October 18th 2011, 09:36 AM
Lolyta

Prove that By is a convex set

Hello.
Definition: A is convex if and only if $\forall{x_1,x_2\in{A}}$ the segment which unite them $(\lambda x_1 + (1-\lambda)x_2 ) \in{A}; 0\leq{\lambda\leq{1}}$ .
So, let consider the set of vectors $By$ where B is a n x m matrix and $y = (y_1,y_2,...,y_m): y_i\geq{0}$ is an m-vector : $y_1+y_2+...+y_m=1$ , prove that By is a convex set.

Thanks a lot.
October 18th 2011, 10:10 AM
girdav

Re: Prove that By is a convex set

What did you try?
October 18th 2011, 10:45 AM
Lolyta

Re: Prove that By is a convex set

Quote:

Originally Posted by girdav

What did you try?

Ok,the main problem is that considering $x_1,x_2\in{By}$ , I don't know how $x_1,x_2$ are. I don't know if you understand me.(Worried)
October 18th 2011, 10:51 AM
girdav

Re: Prove that By is a convex set

Let $v_1$ and $v_2$ in this set. We can write $v_1=Bx$ and $v_2=By$ , where $x=(x_1,\ldots,x_m)$ and $y=(y_1,\ldots,y_n)$ are such that $x_j,y_j\geq 0$ for all $j$ and $\sum_{j=1}^mx_j=\sum_{j=1}^mx_j=1$ . Take $\alpha\in\left[0,1\right]$ . Then $\alpha v_1+(1-\alpha)v_2=\alpha Bx+(1-\alpha)By=B(\alpha x+(1-\alpha)y)$ . What about $\alpha x+(1-\alpha)y$ ?
October 18th 2011, 11:13 AM
Lolyta

Re: Prove that By is a convex set

wow! http://latex.codecogs.com/png.latex?...1-%5Calpha%29y = $\alpha\sum_{j=1}^mx_j + (1-\alpha)\sum_{j=1}^yx_j= \alpha + 1 - \alpha = 1$ which implies that http://latex.codecogs.com/png.latex?...1-%5Calpha%29y is in By.

Thank you!!!!(Itwasntme)

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