Vector space axiom - Linear algebra
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10 Terms
Addition is closed
𝑢 + 𝑣 ∈ 𝑉 for all 𝑢, 𝑣 ∈ 𝑉
Addition is commutative
𝑢 + 𝑣 = 𝑣 + 𝑢 for all 𝑢, 𝑣 ∈ 𝑉
Addition is associative
𝑢 + (𝑣 + 𝑤) = (𝑢 + 𝑣) + 𝑤 for all 𝑢, 𝑣, 𝑤 ∈ 𝑉
V contains a zero element
𝑢 + 0 = 𝑢 for all 𝑢 ∈ 𝑉
Each 𝑢 in 𝑉 has a negative
𝑢 ∈ 𝑉, there exists an element −𝑢 of 𝑉 such that 𝑢 + (−𝑢) = 0
Scalar multiplication is closed
𝑘𝑢 ∈ 𝑉 for all 𝑘 ∈ ℝ & 𝑢 ∈ 𝑉
Addition & scalar multiplication satisfy the first distributive law
𝑘(𝑢 + 𝑣) = 𝑘𝑢 + 𝑘𝑣 for all 𝑘 ∈ ℝ & 𝑢, 𝑣 ∈ 𝑉
Addition & scalar multiplication satisfy the second distributive law
(𝑘 + 𝑙)𝑢 = 𝑘𝑢 + 𝑙𝑢 for all 𝑘, 𝑙 ∈ ℝ & 𝑢 ∈ 𝑉
Ninth axiom
(𝑘𝑙)𝑢 = 𝑘(𝑙𝑢) for all 𝑘, 𝑙 ∈ ℝ & 𝑢 ∈ 𝑉
The real number 1
1𝑢 = 𝑢 for all 𝑢 ∈ 𝑉
Note 
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