= Linearity = Linear algebra is fundamentally algebra that can be applied to any linear space. <> ---- == Axioms == A linear space obeys these axioms. 1. Associativity of addition i. ''(a + b) + c = a + (b + c)'' 2. Commutatibility of addition i. ''a + b = b + a'' 3. There is some ''0'' space that has an additive identity property i. ''0 + a = a'' 4. There is some ''-a'' space for every ''a'' space that has an additive identity property i. ''a + (-a) = 0'' 5. Commutatibility of scalar multiplication i. if ''a'' and ''b'' are scalars while ''c'' is a space i. ''a(bc) = (ab)c'' 6. Identity of scalar multiplication i. ''1a = a'' 7. Distributivity of scalar multiplication i. if ''a'' is a scalar while ''b'' and ''c'' are spaces i. ''a(b + c) = ab + ac'' 7. Distributivity of space multiplication i. if ''a'' and ''b'' are scalars while ''c'' is a space i. ''(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