# standard basis

If $R$ is a division ring, then the direct sum (http://planetmath.org/DirectSum) of $n$ copies of $R$,

 $R^{n}=R\oplus\cdots\oplus R\text{ (n times),}$

is a vector space.

The standard basis for $R^{n}$ consists of $n$ elements

 $e_{1}=(1,0,\ldots,0),\quad e_{2}=(0,1,0,\ldots,0),\quad\ldots\quad e_{n}=(0,% \ldots,0,1)$

where each $e_{i}$ has $1$ for its $i$th component and $0$ for every other component. The $e_{i}$ are called the standard basis vectors.

Title standard basis StandardBasis 2013-03-22 14:20:07 2013-03-22 14:20:07 Mathprof (13753) Mathprof (13753) 9 Mathprof (13753) Definition msc 15A03 BasalUnits standard basis vectors