Vector space

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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.


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