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)

Author: mathtuition88

2 thoughts on “How to Remember the 8 Vector Space Axioms”

  1. 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

    Liked by 1 person

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