Vector space

A set closed under addition and scalar multiplication. One example is $LaTeX: \textstyle \mathbb{R}^n,$ where addition is the usual coordinate-wise addition, and scalar multiplication is $LaTeX: \textstyle t(x_1,\dots,x_n) = (tx_1,\dots,tx_n).$ Another vector space is the set of all $LaTeX: \textstyle m \mbox{x} n$ matrices. If $LaTeX: A$ and $LaTeX: B$ are two matrices (of the same size), so is $LaTeX: A+B.$ Also, $LaTeX: tA$ is a matrix for any scalar, $LaTeX: \textstyle t \in \mathbb{R.}$ Another vector space is the set of all functions with domain $LaTeX: X$ and range in $LaTeX: \textstyle \mbox{R}^n.$ If $LaTeX: \textstyle f \mbox{ and } g$ are two such functions, so are $LaTeX: \textstyle f+g \mbox{ and } tf \mbox{ for all } t \in \mbox{R.}$ Note that a vector space must have a zero since we can set $LaTeX: t=0.$