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Revision difference : standard basis |
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Version 5 |
| \PMlinkescapeword{component} |
\PMlinkescapeword{component} |
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| If $R$ is a division ring, then the \PMlinkname{direct sum}{DirectSum} of $n$ copies of $R$, |
If $R$ is a division ring, then the \PMlinkname{direct sum}{DirectSum} of $n$ copies of $R$, |
| \[ R^n = R \oplus\dotsb\oplus R\text{ (n times),}\] |
\[ R^n = R \oplus\dotsb\oplus R\text{ (n times),}\] |
| is a vector space. |
is a vector space. |
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| The \emph{standard basis for $R^n$} consists of $n$ elements |
The \emph{standard basis for $R^n$} consists of $n$ elements |
| \[ e_1 = (1,0,\dotsc ,0), \quad e_2 = (0,1,0,\dotsc ,0),\quad \dotsc \quad e_n = (0,\dotsc ,0,1) \] |
\[ e_1 = (1,0,\dotsc ,0), e_2 = (0,1,0,\dotsc ,0),\dotsc , e_n = (0,\dotsc ,0,1) \] |
| where each $e_i$ has $1$ for its $i$th component and $0$ for every other component. |
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. |
| The $e_i$ are called the \emph{standard basis vectors}. |
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