Algebraic Properties of Vectors

Vector addition and scalar multiplication satisfy a rich set of algebraic properties. These properties hold for any vectors in $\mathbb{R}^m$ and any scalars, and they follow directly from the fact that real number arithmetic satisfies these rules entry-by-entry.

The most important properties are: commutativity ($\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$), associativity ($(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$), and distributivity ($c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$).

These 8 properties — when abstracted from $\mathbb{R}^m$ — define what a vector space is. Any system satisfying these axioms behaves like vectors, even if its elements look nothing like number tuples (e.g., polynomials, matrices, continuous functions).