Linear independence

Let $ \mathbf{a}_1, \ldots, \mathbf{a}_n$ be column vectors in  $ \mathbb{R}^m$. Then we have the homogeneous equation

$\displaystyle x_1 \mathbf{a}_{1} + \cdots + x_n \mathbf{a}_{n} = \mathbf{0}$ (10)

in the form of vector equation. If the above homogeneous equation has only the trivial solution  $ x_1 = \cdots = x_n = 0$, then the vectors  $ \mathbf{a}_1, \ldots, \mathbf{a}_n$ are said to be linearly independent; otherwise, they are linearly dependent.

EXAMPLES 2. Determine whether the vectors $ \mathbf{v}_1$, $ \mathbf{v}_2$ and $ \mathbf{v}_3$ are linearly independent or not in each of the following.

  1. $ \mathbf{v}_1 = \begin{bmatrix}
1 \\
2 \\
3
\end{bmatrix}$, $ \mathbf{v}_2 = \begin{bmatrix}
4 \\
5 \\
6
\end{bmatrix}$, and $ \mathbf{v}_3 = \begin{bmatrix}
2 \\
1 \\
0
\end{bmatrix}$
  2. $ \mathbf{v}_1 = \begin{bmatrix}
0 \\
1 \\
5
\end{bmatrix}$, $ \mathbf{v}_2 = \begin{bmatrix}
1 \\
2 \\
8
\end{bmatrix}$, and $ \mathbf{v}_3 = \begin{bmatrix}
4 \\
-1 \\
0
\end{bmatrix}$



Department of Mathematics
Last modified: 2005-10-21