Linearity

Linear algebra is fundamentally algebra that can be applied to any linear space.

Contents

  1. Linearity
    1. Axioms

Axioms

A linear space obeys these axioms.

  1. Associativity of addition

    1. (a + b) + c = a + (b + c)

  2. Commutatibility of addition

    1. a + b = b + a

  3. There is some 0 space that has an additive identity property

    1. 0 + a = a

  4. There is some -a space for every a space that has an additive identity property

    1. a + (-a) = 0

  5. Commutatibility of scalar multiplication

    1. if a and b are scalars while c is a space

    2. a(bc) = (ab)c

  6. Identity of scalar multiplication

    1. 1a = a

  7. Distributivity of scalar multiplication

    1. if a is a scalar while b and c are spaces

    2. a(b + c) = ab + ac

  8. Distributivity of space multiplication

    1. if a and b are scalars while c is a space

    2. (a + b)c = ac + bc

Vectors, matrices, and subspaces all obey these axioms, laying the foundation for linear algebra.

Functions also obey these axioms.

CategoryRicottone

LinearAlgebra/Linearity (last edited 2024-01-31 16:40:10 by DominicRicottone)