[선형대수학] 순서 기저 (Ordered Basis)

$\underline{Def}$

Let $V$ be a finite dimensional vector space.

An ordered basis for $V$ is  a finite sequence of linear independent vectors in $V$ that spans $V$.

 

$\underline{Ex}$

In $\mathbb{F}^{3},\, \{ e_{1},e_{2},e_{3} \}$ is an ordered basis for $\mathbb{F}^{3}$.

Also, $\{e_{2},e_{1},e_{3} \}$ is another ordered basis for $\mathbb{F}^{3}$.

 

$\underline{Rmk}$

For a given basis $\beta=\{v_{1},\cdots,v_{n}\}$ of order $n$, there are precisely $n\!$ distinct ordered bases.

 

$\underline{Def}$

$\{e_{1},\cdots,e_{n}\}$ is called the standard ordered basis for $\mathbb{F}^{n}$.

$\{1,x,\cdots,x^{n}\}$ is called the standard ordered basis for $P_{n}(\mathbb{F})$.