Superset of Linearly Dependent Set

Theorem

Any set containing a linearly dependent set is also linearly dependent.

Proof

Suppose $\left\{{a_1, a_2, \ldots, a_n}\right\}$ is a linearly independent set.

Then by Subset of Linearly Independent Set, any subset $\left\{{b_1, b_2, \ldots, b_m}\right\}$ of $\left\{{a_1, a_2, \ldots, a_n}\right\}$ must itself be linearly independent.

Thus if $\left\{{b_1, b_2, \ldots, b_m}\right\}$ is linearly dependent, then so must $\left\{{a_1, a_2, \ldots, a_n}\right\}$ be.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 27$