Vector Space has a total of 8 Axioms, most of which are common-sense, but can still pose a challenge for memorizing by heart.

I created a mnemonic “**MAD**” which helps to remember them.

**M for Multiplicative Axioms:**

- (Scalar Multiplication identity)
- (Associativity of Scalar Multiplication)

**A for Additive Axioms: (Note that these are precisely the axioms for an abelian group)**

- (Commutativity)
- (Associativity for Vector Addition)
- (Existence of Additive Inverse)
- (Additive Identity)

**D for Distributive Axioms:**

- (Distributivity of vector sums)
- (Distributivity of scalar sums)

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*Related*

For Group, “CAN I?” is the memorize trick: 群

C = Closure

A = Associalize

N = Neutral element (Identity, e)

I = Inverse

Less 50% discount => “CA” for Semi-Group 半群

“CAN” => Mono’I’d (no ‘I’: Group with no Inverse) 么群 Monoid

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sorry for spelling, A = Associativity

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