Exists and exist follow the ordinary convention for verbs: one is singular and the other is plural. Where mathematical usage differs from ordinary usage is in the way singular and plural are indicated in the subject that follows, and an implied “for all” later if the subject is plural.
Exists is singular:
There exists s'' in S such that s''(s'm – sm') = 0.
Or, spelled out more explicitly:
There exists a number s'' in the set S, such that s''(s'm – sm') = 0.
Exist is plural:
There exist αi in I such that xn = Σ αi xi.
This is where the mathematical usage differs from ordinary usage. Spelled out explicitly, this would be:
There exist numbers α1, α2, α3, etc., which are elements of the set I, such that for each subscript, if we refer to the subscript as i, then xn = Σ αi xi.
The convention is tricky for a beginner to understand because it depends on your knowing that i is commonly used as a variable that will stand for multiple subscripts. The plural form of “exist” is actually a helpful clue. However, you must understand the implied “for each” later in the sentence. It’s implied by the fact that the sentence uses i to stand for multiple values.