Linearly Dependent Vectors

$n$ Vectors ${\bf X}_1$, ${\bf X}_2$, ..., ${\bf X}_n$ are linearly dependent Iff there exist Scalars $c_1$, $c_2$, ..., $c_n$, not all zero, such that

$$c_i{\bf X}_i = 0,$$ (1)

where Einstein Summation is used and $i=1$, ..., $n$. If no such Scalars exist, then the vectors are said to be linearly independent. In order to satisfy the Criterion for linear dependence,

$$c_1\left[{\matrix{x_{11}\cr x_{21}\cr \vdots\cr x_{n1}\cr}}\... ...}\cr}}\right]=\left[{\matrix{0\cr 0\cr \vdots\cr 0\cr}}\right]$$ (2)

$$\left[{\matrix{ x_{11} & x_{12} & \cdots & x_{1n}\cr x_{21... ...}}\right] = \left[{\matrix{0\cr 0\cr \vdots\cr 0\cr}}\right].$$ (3)

In order for this Matrix equation to have a nontrivial solution, the Determinant must be 0, so the Vectors are linearly dependent if

$$\left\vert\matrix{x_{11} & x_{12} & \cdots & x_{1n}\cr x_{2... ...\vdots\cr x_{n1} & x_{n2} & \cdots & x_{nn}\cr}\right\vert=0,$$ (4)

and linearly independent otherwise.

Let ${\bf p}$ and ${\bf q}$ be $n$-D Vectors. Then the following three conditions are equivalent (Gray 1993).

1. ${\bf p}$ and ${\bf q}$ are linearly dependent.

2. $\left\vert\matrix{{\bf p}\cdot{\bf p} & {\bf p}\cdot{\bf q}\cr {\bf q}\cdot{\bf p} & {\bf q}\cdot{\bf q}\cr}\right\vert=0$.

3. The $2\times n$ Matrix $\left[{\matrix{{\bf p}\cr {\bf q}\cr}}\right]$ has rank less than two.

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 186-187, 1993.