| Version 25 |
Version 24 |
| Let $V$ be a vector space over a |
Let $V$ be a vector space over a |
|
field $F$. We say that $v_1,\ldots, v_k\in V$ are \emph{linearly dependent} if there exist scalars $\lambda_1,\ldots, \lambda_k\in F$, not all zero, such that
|
field $F$. We say that $v_1,\ldots, v_k\in V$ are linearly dependent if there exist scalars $\lambda_1,\ldots, \lambda_k\in F$, not all zero, such that
|
| \[ |
\[ |
| \lambda_1 v_1+ \cdots +\lambda_k v_k = 0 . |
\lambda_1 v_1+ \cdots +\lambda_k v_k = 0 . |
| \] |
\] |
| If no such scalars exist, then we say that the vectors are \emph{linearly independent}. |
If no such scalars exist, then we say that the vectors are \emph{linearly independent}. |
| More generally, we say that a (possibly infinite) subset $S\subset V$ is linearly independent if all finite subsets of $S$ are linearly independent. |
More generally, we say that a (possibly infinite) subset $S\subset V$ is linearly independent if all finite subsets of $S$ are linearly independent. |
|
|
| In the case of two vectors, linear dependence means that one of the |
In the case of two vectors, linear dependence means that one of the |
| vectors is a scalar multiple of the other. As an alternate |
vectors is a scalar multiple of the other. As an alternate |
| characterization of dependence, we also have the following. |
characterization of dependence, we also have the following. |
| \begin{proposition} |
\begin{proposition} |
| Let $S\subset V$ be a subset of a vector space. Then, $S$ is |
Let $S\subset V$ be a subset of a vector space. Then, $S$ is |
| linearly dependent if and only if there exists a $v\in S$ such that |
linearly dependent if and only if there exists a $v\in S$ such that |
| $v$ can be expressed as a linear combination of the vectors in the |
$v$ can be expressed as a linear combination of the vectors in the |
| set $S\backslash \{v\}$ (\PMlinkname{all the vectors in $S$ other |
set $S\backslash \{v\}$ (\PMlinkname{all the vectors in $S$ other |
| than $v$}{SetDifference}). |
than $v$}{SetDifference}). |
| \end{proposition} |
\end{proposition} |
