Problem:
Solution:
Part a is obvious. The given formula is the parametric line and by definition it is in the convex set C.
Part a is the basis of the hypothesis. Consider it is true for n=k, we have
k+1∑i=1ti(xiyi)=k∑i=1ti(xiyi)+tk+1(xk+1yk+1)=(k∑i=1ti)k∑i=1tik∑i=1ti(xiyi)+tk+1(xk+1yk+1)
The inner sum is actually a point in C because it is a convex combination.The outer sum is also a point in C because that is also a convex combination!
Solution:
Part a is obvious. The given formula is the parametric line and by definition it is in the convex set C.
Part a is the basis of the hypothesis. Consider it is true for n=k, we have
k+1∑i=1ti(xiyi)=k∑i=1ti(xiyi)+tk+1(xk+1yk+1)=(k∑i=1ti)k∑i=1tik∑i=1ti(xiyi)+tk+1(xk+1yk+1)
The inner sum is actually a point in C because it is a convex combination.The outer sum is also a point in C because that is also a convex combination!
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