Zero Vector is Linearly Dependent

Theorem

Let $G$ be a group whose identity is $e$.

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\left({G, +_G, \circ}\right)_R$ be a unitary $R$-module.

Then the singleton set $\left\{{e}\right\}$ consisting of the zero vector is linearly dependent.

Proof

By Scalar Product with Identity we have:

$\forall \lambda \in R: \lambda \circ e = e$

Hence the result by definition of linearly dependent.

$\blacksquare$

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.4$