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Linear algebra is fundamentally algebra that can be applied to any linear space. Linear algebra is algebra that can be applied to any linear space to study [[LinearAlgebra/LinearMapping|linear mappings]]. It is helpful to then define '''linearity'''.
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 2. Commutatibility of addition  2. Commutability of addition
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 5. Commutatibility of scalar multiplication  5. Commutability of scalar multiplication
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Functions also obey these axioms. Functions and [[Calculus/Derivative|derivatives]] are also [[LinearAlgebra/LinearMapping|transformations]], often in polynomial vector space (''P,,n,,''). For example, the transformation ''T'' representing differentiation with respect to ''x'' in the domain ''P,,3,,'' is formalized as ''T: P,,3,, -> P,,2,,''.

Linearity

Linear algebra is algebra that can be applied to any linear space to study linear mappings. It is helpful to then define linearity.

Contents

  1. Linearity
    1. Axioms

Axioms

A linear space obeys these axioms.

  1. Associativity of addition

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

  2. Commutability 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. Commutability 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 and derivatives are also transformations, often in polynomial vector space (Pn). For example, the transformation T representing differentiation with respect to x in the domain P3 is formalized as T: P3 -> P2.

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LinearAlgebra/Linearity (last edited 2026-01-27 01:06:14 by DominicRicottone)