Superset of Linearly Dependent Set is Linearly Dependent

Theorem

Let $S$ be a set of elements of a unitary module.

Let $S$ contain a subset $T$ which is linearly dependent.

Then $S$ is also linearly dependent.

Proof

Let $S$ be a linearly independent set.

Let $T$ be a subset of $S$.

By Subset of Linearly Independent Set is Linearly Independent, $T$ is also linearly independent.

Thus by Proof by Contraposition, if $T$ is linearly dependent, then so must $S$ be.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases