Linear Dependence

A subset $S\subseteq V$ is linearly dependent if there exists a finite number of distinct vectors $u_1,\dots ,u_n\in S$ and scalars $a_1,\dots ,a_n$ not all zero such that $a_1{u}_1+\cdots +a_n{u}_n=0$

Facts

1. The empty set is linearly independent (because linear dependent sets must be nonempty)
2. A set containing one nonzero vector is linear independent
3. A set is linearly independent iff the only representations of 0 as linear combinations of its vectors are trivial representations (all coeffs = 0)

Theorems

• Let $S_1\subseteq S_2\subseteq V$. If $S_1$ is linearly dependent, then $S_2$ is linearly dependent (Theorem 1.6)
  • Contrapositive: If $S_2$ is linearly independent, then $S_1 is linearly independent
• Let $S$ be a linearly independent subset of $V$. Let $v\in V\backslash S$. Then $S\cup \{v\}$ is linearly dependent iff $v\in \text{span}(s)$
  • Intuition: If $S$ is a generating set for a subspace, and no subset of $S$ is still a generating set, then $S$ must be linearly independent