Linearity

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

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 (P_n). For example, the transformation T representing differentiation with respect to x in the domain P₃ is formalized as T: P₃ -> P₂.

CategoryRicottone