Linearly Dependent

Last updated Nov 1, 2022

Let $V$ be a Vector Space on Field $F$. Let $S \subset V$. $S$ is Linearly Dependent if there exists a finite subset ${\mathbf{a}{i}}{i=1}^{n}$ for some $n \in \mathbb{N}$ and constants $c_{1}, \dots, c_{n} \in F$ so that

1. $c_{1} \mathbf{a}{1} + \cdots + c{n} \mathbf{a}_{n} = \mathbf{0}$
2. $\exists i \in [n]$ so that $c_{i} \neq 0$

Let $V$ be a Vector Space on Field $F$. Let $\mathbf{a}{1}, \dots, \mathbf{a}{n} \in V$ for some $n \in \mathbb{N}$. $\mathbf{a}{1}, \dots, \mathbf{a}{n}$ are Linearly Dependent if there exist constants $c_{1}, \dots, c_{n} \in F$ so that

1. $c_{1} \mathbf{a}{1} + \cdots + c{n} \mathbf{a}_{n} = \mathbf{0}$
2. $\exists i \in [n]$ so that $c_{i} \neq 0$

• We call the appropriate constants nontrivial, in that just setting them all to $0$ is the trivial choice.
• In defiition 2, our vectors may be the same. If $\mathbf{a}{i} = \mathbf{a}{j}$ for some $i, j \in [n]$, $i \neq j$, then $\mathbf{a}{i} + (-1)\mathbf{a}{j} = \mathbf{0}$.