## How to Remember the 8 Vector Space Axioms

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:

1. $1x=x$ (Scalar Multiplication identity)
2. $(ab)x=a(bx)$ (Associativity of Scalar Multiplication)

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

1. $x+y=y+x$ (Commutativity)
2. $(x+y)+z=x+(y+z)$ (Associativity for Vector Addition)
3. $x+(-x)=0$ (Existence of Additive Inverse)
4. $x+0=0+x=0$ (Additive Identity)

D for Distributive Axioms:

1. $a(x+y)=ax+ay$ (Distributivity of vector sums)
2. $(a+b)x=ax+bx$ (Distributivity of scalar sums)

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### 2 Responses to How to Remember the 8 Vector Space Axioms

1. Couscous says:

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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2. Couscous says:

sorry for spelling, A = Associativity

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