standard basis
StandardBasis
2004-04-26 03:36:25
2007-10-31 09:48:45
Definition
basis
standard basis vectors
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If $R$ is a division ring, then the \PMlinkname{direct sum}{DirectSum} of $n$ copies of $R$,
\[ R^n = R \oplus\dotsb\oplus R\text{ (n times),}\]
is a vector space.
The \emph{standard basis for $R^n$} consists of $n$ elements
\[ e_1 = (1,0,\dotsc ,0), \quad e_2 = (0,1,0,\dotsc ,0),\quad \dotsc \quad e_n = (0,\dotsc ,0,1) \]
where each $e_i$ has $1$ for its $i$th component and $0$ for every other component.
The $e_i$ are called the \emph{standard basis vectors}.