Стандартний базис

Стандартний базис чи канонічний базис в лінійній алгебрі — спеціальний базис для певного векторного простору, котрий в цьому просторі внаслідок своєї конструкції та форми виділяється серед інших базисів цього векторного простору.

Визначення

Серед найбільш відомих та найуживаніших стандартних векторних просторів $\mathbb {R} ^{n}$ з $n\in \mathbb {N}$ , елементами яких є всі $n$ -кортежі дійсних чисел з-поміж можливих базисів в цих $\mathbb {R} ^{n}$ виділяють ті, відносно яких координати вектора збігаються з компонентами кортежів $e_{1},\ldots ,e_{n}$ таким чином

{\begin{matrix}e_{1}&=&(1,0,0,\ldots ,0),\\e_{2}&=&(0,1,0,\ldots ,0),\\&\vdots &\\e_{n}&=&(0,0,0,\ldots ,1)\end{matrix}}

Цей базис називають стандартним в $\mathbb {R} ^{n}$ . Стандартний базис є ортогональним та ортонормованим.

Приклади

Стандартний базис в $\mathbb {R} ^{2}$ складають вектори

e_{1}={\begin{pmatrix}1\\0\end{pmatrix}},\quad e_{2}={\begin{pmatrix}0\\1\end{pmatrix}}

В тривимірному векторному просторі $\mathbb {R} ^{3}$ три вектори стандартного базису :

\mathbf {i} =e_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},\quad \mathbf {j} =e_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},\quad \mathbf {k} =e_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}

Посилання

Ryan, Patrick J. (1986). Euclidean and non-Euclidean geometry: an analytical approach. Cambridge; New York: Cambridge University Press. ISBN 0-521-27635-7. (с. 198) (англ.)

Schneider, Philip J.; Eberly, David H. (2003). Geometric tools for computer graphics. Amsterdam; Boston: Morgan Kaufmann Publishers. ISBN 1-55860-594-0. (с. 112) (англ.)

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