Definition of linear independence/dependence and examples

  A set of $n$ vectors $\{ \mathbf{x}_{1}, \mathbf{x}_{2}, \cdots, \mathbf{x}_{n} \}$ is said to be linearly independent iff the equation holds only when all the coeffients $c_{1}, c_{2}, \cdots, c_{n}$ are equal to zero, i.e., . On the other hand, a set of $n$ vectors $\{ \mathbf{y}_{1}, \mathbf{y}_{2}, \cdots, \mathbf{y}_{n} \}$ is said to be linearly dependent iff the equation holds and there exists non-zero coeffient $d_{i}$ for any $i$, i.e., ($i = 1, 2, \cdots, n$.)
  Below are a brief supplement and examples.

Explanation

  Acoording to the above definition, if a set of vectors $\{ \mathbf{y}_{1}, \mathbf{y}_{2}, \cdots, \mathbf{y}_{n} \}$ is linearly dependent, at least one vector in the vectors can be represented by a linear combination of the remaining vectors. For example, if $d_{i} \neq 0$, $\mathbf{y}_{i}$ can be represented as . On the other hand, if the vectors are linearly independent, such expression is not permitted.
  Therefore, a set of vectors is said to be linearly dependent when at least one vector in the vectors can be represented by a linear combination of the remaining vectors. On the other hand, a set of vectors is said to be linearly independent when any vector can not be represented by a linear combination of the remaining vectors.   A set of vectors is linearly independent, because if the equation holds, then .   Let us investigate whether a set of vectors is linearly independent.
  If the equation holds, the coefficients satisfy . The solution is , which is not non-zero. In this way, there exists non-zero coefficient satisfying $(1)$. Therefore, the set of vectors $ \{ \mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3} \} $ is linearly dependent. Therefore at least one vector is represented by a linear combination of the other vectors. For example, $\mathbf{x}_{3}$ is represented as .   Let us investigate a set of vectors is linearly independent. If the equation holds, then the coeffients $c_{1},c_{2},c_{3}$, satisfy . From this, we have . Therefore $ \{\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3} \} $ is linearly independent.