Consider a linear system equation

*, *where

* * is in general an

matrix and

and

are respectively an

and an

vector. Then Farkas's lemma states that one but not both of the following two statements is true: (1) there is a vector

solving the equation; or (2) there is a vector

satisfying

and

, where

is the zero vector. Geometrically, this is equivalent to saying that: (1)

is in the cone

spanned by

; or (2)

is not in

. To show the equivalence is simple. The first statement requires directly that

is in

, so that

. If (1) does not hold, then

is not in

; therefore, we will find a separating hyperplane between

and

. In that case, we can find a vector

whose angles with each row of

are at most

. Because such a

is included in

, its angle with

must be greater than

. In order to understand this lemma intuitively, it is more or less sufficient to understand the two-dimensional case.