Fundamental Properties of Vector Algebra

Theorem

1. $\overrightarrow{x}$ + $\overrightarrow{y}$ $\in$ ${\mathbb{R}}^n$ | ${\mathbb{R}}^n$ is closed under addition
2. $\overrightarrow{x}$ + $\overrightarrow{y}$ = $\overrightarrow{y}$ + $\overrightarrow{x}$ | Addition is commutative
3. ($\overrightarrow{x}$ + $\overrightarrow{y}$ ) + $\overrightarrow{w}$ = $\overrightarrow{x}$ + ($\overrightarrow{y}$ + $\overrightarrow{w}$) | Addition is associative
4. c $\overrightarrow{x}$ $\in$ ${\mathbb{R}}^n$ | ${\mathbb{R}}^n$ is closed under scalar multiplication
5. c(d $\overrightarrow{x}$) = (cd) $\overrightarrow{x}$ | Scalar Multiplication is associative
6. (c + d) $\overrightarrow{x}$ = c $\overrightarrow{x}$ + d $\overrightarrow{x}$ | Distributive Law
7. c($\overrightarrow{x}$ + $\overrightarrow{y}$) = c$\overrightarrow{x}$ + c$\overrightarrow{y}$ | Distributive Law

Vector Algebra behaves as expected, but it isn’t always nice for example AB $\ne$ BA