| Version 17 |
Version 16 |
| Let $V$ be a vector space over a |
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 |
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 condtions holds: |
linearly independent if the following condtions holds: |
| \lambda_1\vec{v}_1+ \lambda_2 \vec{v}_2 + ~\cdots ~ +\lambda_n \vec{v}_n = 0$ \mbox{ implies } |
\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 |
\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 |
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. |
independent. |
| In the case of two vectors, linear independence means that one of these vectors is not a scalar multiple of the other. |
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 |
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 |
set lies in the linear span of the other vectors in the set |
