Vectors
,
, ...,
are linearly dependent Iff there
exist Scalars
,
, ...,
, not all zero, such that
![\begin{displaymath}
c_i{\bf X}_i = 0,
\end{displaymath}](l2_365.gif) |
(1) |
where Einstein Summation is used and
, ...,
. If no such Scalars exist,
then the vectors are said to be linearly independent. In order to satisfy the Criterion for linear dependence,
![\begin{displaymath}
c_1\left[{\matrix{x_{11}\cr x_{21}\cr \vdots\cr x_{n1}\cr}}\...
...}\cr}}\right]=\left[{\matrix{0\cr 0\cr \vdots\cr 0\cr}}\right]
\end{displaymath}](l2_366.gif) |
(2) |
![\begin{displaymath}
\left[{\matrix{
x_{11} & x_{12} & \cdots & x_{1n}\cr
x_{21...
...}}\right]
= \left[{\matrix{0\cr 0\cr \vdots\cr 0\cr}}\right].
\end{displaymath}](l2_367.gif) |
(3) |
In order for this Matrix equation to have a nontrivial solution, the Determinant must be 0, so the
Vectors are linearly dependent if
![\begin{displaymath}
\left\vert\matrix{x_{11} & x_{12} & \cdots & x_{1n}\cr
x_{2...
...\vdots\cr
x_{n1} & x_{n2} & \cdots & x_{nn}\cr}\right\vert=0,
\end{displaymath}](l2_368.gif) |
(4) |
and linearly independent otherwise.
Let
and
be
-D Vectors. Then the following three conditions are equivalent
(Gray 1993).
- 1.
and
are linearly dependent.
- 2.
.
- 3. The
Matrix
has rank less than two.
References
Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 186-187, 1993.
© 1996-9 Eric W. Weisstein
1999-05-25