Definitions
An example from linear algebra
Definition: A set \(\{v_1\ldots,v_n\}\) of elements of a vector space \(V\) is called linearly independent if, for any set \(a_1,\ldots,a_n\) of scalars, if \(\sum_{i=1}^{n} a_{i}v_{i}=0\) then \(a_{i}=0\) for all \(i=1,\ldots, n\). A set of vectors that is not linearly independent is called linearly dependent.
Theorem: The set of vectors \(\{(1,3),(2,2)\}\) is linearly independent in \(\mathbb{R}^{2}\).
Theorem: The set of vectors \(\{(1,1,1),(2,2,2),(1,3,2)\}\) is linearly dependent in \(\mathbb{R}^{3}\).