|
|
|
Viewing Version
22
of
'linear independence'
|
[ view 'linear independence'
|
back to history
]
| Title of object: |
linear independence |
| Canonical Name: |
LinearIndependence |
| Type: |
Definition |
| Created on: |
2001-11-14 15:58:31 |
| Modified on: |
2003-02-03 00:40:40 |
| Classification: |
msc:15A03 |
| Synonyms: |
linear independence=linearly independent |
Revision comment (for changes between this and next version):
| Changes for correction #7816 ('logical phrasing'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $V$ be a vector space over a
field $F$. Then for scalars $\lambda_1,~ \lambda_2,~ \ldots, ~\lambda_n \in F$ the vectors $\vec{v}_1,~ \vec{v}_2,~ \ldots,~ \vec{v}_n \in V$ are
linearly independent if the following condtion holds:
\[
\lambda_1\vec{v}_1+ \lambda_2 \vec{v}_2 + ~\cdots ~ +\lambda_n \vec{v}_n = 0 \mbox{ implies }
~ \lambda_1 = \lambda_2 = ~\ldots~ = \lambda_n=0
\]
Otherwise, if this conditions fails, the vectors are said to be linearly dependent. Furthermore, an infinite set of vectors is linearly independent if all \emph{finite} subsets are linearly
independent.
In the case of two vectors, linear independence means that one of these vectors is not a scalar multiple of the other.
As an alternate characterization of dependence, we have that a set of of vectors is linearly dependent if and only if some vector in the
set lies in the linear span of the other vectors in the set. |
|
|
|
|
|
