向量之基底

原理

在三维向量中,任意向量可以由基底(三个线性无关的向量)表示.

向量定义

定义 2.11 线性无关. 假设

$a_1e_1 + ⋯ + a_ne_n = 0 $

只有在 $a_1 = ⋯ = a_n = 0$ 时成立，那么向量 $\{e_1, e_2, ..., e_n\}$ 是线性无关的。如果任何 $a_i$ 不为零，那么这些向量是线性相关的。其中一个向量是其他向量的组合。

Definition 2.11. A set of $n$ vectors $\{e_1, e_2, ..., e_n\}$ is linearly independent if there do not exist real numbers $\{a_1, a_2, ..., a_n\}$, where at least one of the $a_i$ is not zero, such that:

$a_1e_1 + ⋯ + a_ne_n = 0$ (2.40)

Otherwise, the set $\{e_1, e_2, ..., e_n\}$ is called linearly dependent.

An n-dimensional vector space is one that can be generated by a set of $n$ linearly independent vectors. Such a generating set is called a basis, whose formal definition follows.

Definition 2.12. A basis $B3$ for a vector space $V$ is a set of n linearly independent vectors B3 $\{e_1, e_2, ..., e_n\}$ for which, given any element $P$ in $V $, there exist real numbers $\{a_1, a_2, ..., a_n\}$ such that

$P = a_1e_1 + ⋯ + a_ne_n​$ (2.41)