ua: Vectors

A matrix with only one column is called acolumn vector, or simply a vector.

It means that a vector in $\mathbb{R}^n$ is represented as an ordered list of n real numbers, written as $(u_1, ..., u_n)$

The vector whose entries are all zero is called thezero vectorand is denoted by0.
(The number of entries in0will be clear from the context.)

Vector addition is an operation on vectors in $\mathbb{R}^n$ where corresponding components are added, and it satisfies $u+v=v+u$.

The associative property of vector addition states that $(u+v)+w=u+(v+w)$ for all $u,v,w\in\mathbb{R}^n$.

The additive identity is the zero vector $0$, which satisfies $u+0=0+u=u$ for all $u\in\mathbb{R}^n$.

The additive inverse of a vector $u$ is $-u$, and it satisfies $u+(-u)=0$.

For any scalar $c$ and vectors $u,v\in\mathbb{R}^n$, scalar multiplication satisfies $c(u+v)=cu+cv$.

For any scalars $c,d$ and vector $u\in\mathbb{R}^n$, $(c+d)u=cu+du$.

For any scalars $c,d$ and vector $u\in\mathbb{R}^n$, scalar multiplication satisfies $c(du)=(cd)u$.

The multiplicative identity property states that $1u=u$ for all $u\in\mathbb{R}^n$.