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 $\struct {G, +_G, \circ}_R$ be a unitary $R$-module.

Then the singleton set $\set e$ 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

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 4$. Vector Spaces