Question

Exercise 17: Let X C Rn be non-empty and let A be an n × n matrix. Show that Alco (X)-co (A Here co means convex hull. )

Please show all work so I can gain a better understanding. Thank you!

(Let X ⊂ R n be non-empty and let A be an n×n matrix. Show that A[co (X)] = co (A[X]). Here co means convex hull.)

Answer 1

Let begin by finding out how an element is A[co(X)] looks like.

Let y be in co(X). Then from the definition of convex hull

CLİ .Tİ

where n is a finite number, a_i are in [0,1] and x_i is in X and,

.

ai = 1 に!

Since A a matrix and hence a linear transform on R^n we have that

に!
.

But notice that

に!

is a finite linear combination of elements is A(X) with the additional property that

.

ai = 1 に!

So Ay is an element of co(A(X)).

So we have the following containment.

.

A[co(X))co(A(X)) CO

Now we will prove the reverse containment.

Let v be an element in co(A(X)).

Then according to the definition of convex hull we have that

$v=\sum_{j=1}^kb_jAx_j$, where b_j are in [0,1] , x_j in X and

.

bi = 1

Since A is a linear transformation

$v=\sum_{j=1}^kb_jAx_j=A(\sum_{j=1}^kb_jx_j)$

and clearly

.

br) E co

So v an element of A[co(X)].

So

.

co(A(X)) C Aco(X)

This shows

.