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A vector space is a set V together with operations of vector addition and scalar multiplication that satisfy a number of axioms (properties). These properties include closure under addition and scalar multiplication, commutative and associative laws for addition, existence of additive identity, and existence of additive inverses.

One of the properties of vector spaces is the existence of an additive identity—an element 0 in the vector space such that for any other element, v in V, the following is true: \(v + 0 = v\)

Let's assume that there are two additive identity elements in the vector space, 0 and 0'. That means both of these elements satisfy the additive identity property: 1. \(v + 0 = v\) 2. \(v + 0' = v\)

To prove that 0 and 0' are actually the same element and that the additive identity is unique in the vector space, let's add 0 to the equation involving 0': \(v + 0' = v\) Add 0 to both sides: \(0 + (v + 0') = 0 + v\) By the associative law of addition, we can rearrange the parentheses: \((0 + v) + 0' = v + 0\) Now substitute the additive identity property (1) to get: \(v + 0' = v\) Since we assumed that 0' was also an additive identity, this equation is equivalent to our second assumption. This means that 0' and 0 have the exact same effect on any element v in the vector space. Thus, since 0 and 0' are indistinguishable from one another, only one unique additive identity element exists in the vector space. The element 0 in a vector space is unique.