Vector space axiom - Linear algebra

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10 Terms

1
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Addition is closed

𝑢 + 𝑣 ∈ 𝑉 for all 𝑢, 𝑣 ∈ 𝑉

2
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Addition is commutative

𝑢 + 𝑣 = 𝑣 + 𝑢 for all 𝑢, 𝑣 ∈ 𝑉

3
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Addition is associative

𝑢 + (𝑣 + 𝑤) = (𝑢 + 𝑣) + 𝑤 for all 𝑢, 𝑣, 𝑤 ∈ 𝑉

4
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V contains a zero element

𝑢 + 0 = 𝑢 for all 𝑢 ∈ 𝑉

5
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Each 𝑢 in 𝑉 has a negative

 𝑢 ∈ 𝑉, there exists an element −𝑢 of 𝑉 such that 𝑢 + (−𝑢) = 0

6
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Scalar multiplication is closed

𝑘𝑢 ∈ 𝑉 for all 𝑘 ∈ ℝ & 𝑢 ∈ 𝑉

7
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Addition & scalar multiplication satisfy the first distributive law

𝑘(𝑢 + 𝑣) = 𝑘𝑢 + 𝑘𝑣 for all 𝑘 ∈ ℝ & 𝑢, 𝑣 ∈ 𝑉

8
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Addition & scalar multiplication satisfy the second distributive law

(𝑘 + 𝑙)𝑢 = 𝑘𝑢 + 𝑙𝑢 for all 𝑘, 𝑙 ∈ ℝ & 𝑢 ∈ 𝑉

9
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Ninth axiom

(𝑘𝑙)𝑢 = 𝑘(𝑙𝑢) for all 𝑘, 𝑙 ∈ ℝ & 𝑢 ∈ 𝑉

10
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The real number 1

1𝑢 = 𝑢 for all 𝑢 ∈ 𝑉