“linear independence”的概念、定义、翻译、参考文献-科学参考

linear independence

Computer

A fundamental concept in mathematics. Let
$x_{1}, x_{2}, \dots, x_{n}$
be m-component vectors. These vectors are linearly independent if for some scalars α‎₁, α‎₂,…, α‎_n,
$\sum_{i = 1}^{n} α_{i} x_{i} = 0$
implies
$α_{1} = α_{2} = \dots = α_{n} = 0$
Otherwise the vectors are said to be linearly dependent, i.e. at least one of the vectors can be written as a linear combination of the others. The importance of a linearly independent set of vectors is that, providing there are enough of them, any arbitrary vector can be represented uniquely in terms of them.
A similar concept applies to functions f₁(x), f₂(x),…, f_n(x) defined on an interval [a,b], which are linearly independent if for some scalars α‎₁, α‎₂,…, α‎_n, the condition,
$\begin{array}{l} \sum_{i = 1}^{n} a_{i} f_{i} (x) = 0 \\ for all x in [a, b], \end{array}$
implies
$α_{1} = α_{2} = \dots = α_{n} = 0$

单词	linear independence
释义	linear independence Computer A fundamental concept in mathematics. Let $x_{1}, x_{2}, \dots, x_{n}$ be m-component vectors. These vectors are linearly independent if for some scalars α‎₁, α‎₂,…, α‎_n, $\sum_{i = 1}^{n} α_{i} x_{i} = 0$ implies $α_{1} = α_{2} = \dots = α_{n} = 0$ Otherwise the vectors are said to be linearly dependent, i.e. at least one of the vectors can be written as a linear combination of the others. The importance of a linearly independent set of vectors is that, providing there are enough of them, any arbitrary vector can be represented uniquely in terms of them. A similar concept applies to functions f₁(x), f₂(x),…, f_n(x) defined on an interval [a,b], which are linearly independent if for some scalars α‎₁, α‎₂,…, α‎_n, the condition, $\begin{array}{l} \sum_{i = 1}^{n} a_{i} f_{i} (x) = 0 \\ for all x in [a, b], \end{array}$ implies $α_{1} = α_{2} = \dots = α_{n} = 0$
随便看	caudal vertebrae caudillo Cauer filter cauldron-subsidence causal chain causal diagram causal/evidential decision theory causality causality((in Granger’s sense)) causal law causal nexus causal principle causal reasoning causal regress, infinite causal system causal theory of identity causal theory of knowledge/justification causal theory of meaning causal theory of perception causal uniformity, principle of causa sui causation cause, immanent causes: material, formal, efficient, final causes/reasons