Farkas's Lemma in Two Dimensions
Farkas's Lemma in Two Dimensions
This Demonstration shows a two-dimensional graphical exposition of Farkas's lemma: let be a matrix and be a vector. Then either (1) there is a vector so that and ; or (2) there is a vector satisfying and . As is well known, geometrically this statement is equivalent to saying that is either in the cone spanned by or not. In this Demonstration, and are shown by green and orange lines and the inside of the cone is filled with light blue if (1) is true and with light red if (2) is true.
A
2×2
b
2×1
2×1
x
x≥0
A·x=b
y
A'·y≥0
b'·y<0
b
A
b
x