How to Meet Vector Space Requirements
In linear algebra, a set of elements is termed a vector space when particular requirements are met. For example, let a set consist of vectors u, v, and w. Also let k and l be real numbers, and consider the defined operations of ⊕ and ⊗. The set is a vector space if, under the operation of ⊕, it meets the following requirements:

Closure. u ⊕ v is in the set.

Commutativity. u ⊕ v = v ⊕ u.

Associativity. u ⊕ (v ⊕ w) = (u ⊕ v) ⊕ w.

An identity element 0. u ⊕ 0 = 0 ⊕ u = u for any element u.

An inverse element −u. u ⊕ −u = −u ⊕ u = 0
Under the operation of ⊗, the set is a vector space if it meets the following requirements:

Closure. k ⊗ u is in the set.

Distribution over a vector sum. k ⊗ (u ⊕ v) = k ⊗ u ⊕ k⊗ v.

Distribution over a scalar sum. (k + l) ⊗ u = k ⊗u ⊕ l ⊗ u.

Associativity of a scalar product. k ⊗ (l ⊗ u) = (kl) ⊗ u.

Multiplication by the scalar identity. 1 ⊗ u = u.