Linearly Independent Sets of Vectors

Definition: Let $E$ be a vector space. A subset $A \subseteq E$ is a Linearly Independent set of vectors, if for all $n \in \mathbb{N}$ and for all vectors $x_1, x_2, ..., x_n \in A$, the equation $\lambda_1x_1 + \lambda_2x_2 + ... + \lambda_n x_n = 0$ implies that $\lambda_1 = \lambda_2 = ... = \lambda_n = 0$. A subset $A \subseteq E$ is a Linearly Dependent set of vectors if it is not linearly independent.

For example, in the vector space $\mathbb{R}^3$, the set of vectors $A$ which consists only of $(1, 1, 1)$ and $(1, 2, 1)$ is linearly independent, since the equation:

(1)

\begin{align} \quad \lambda_1 (1, 1, 1) + \lambda_2 (1, 2, 1) = (0, 0, 0) \end{align}

Is equivalent:

(2)

\begin{align} (\lambda_1 + \lambda_2, \lambda_1 + 2\lambda_2, \lambda_1 + \lambda_2) = (0, 0, 0) \end{align}

The first coordinate implies that $\lambda_1 = -\lambda_2$, while the second coordinate implies that $\lambda_1 = -2 \lambda_2$. Thus $\lambda_2 = 2 \lambda_2$ which happens if and only if $\lambda_2 = 0$. But then $\lambda_1 = 0$ too, so $A := \{ (1, 1, 1), (1, 2, 1) \}$ is a linearly independent subset of $\mathbb{R}^3$.

On the other hand, the subset $B := \{ (1, 1, 1), (2, 2, 2) \} \subset \mathbb{R}^3$ is linearly dependent, since for example, the equation:

(3)

\begin{align} \lambda_1 (1, 1, 1) + \lambda_2(2, 2, 2) = (0, 0, 0) \end{align}

Is satisfied for $\lambda_1 = 2$ and $\lambda_2 = 1$, for example.