Let \(U\) and \(V\) be be vector spaces over a field \(\mathbb F\). Linear operators \(T_1,\ldots,T_n:U\to V\) are said to be locally linearly dependent if \(T_1u,\ldots,T_nu\) are linearly dependent for every \(u\in U\). A classical result by Kaplansky asserts that if \(T\) is a linear operator on a vector space \(X\), then \(T\) is algebraic of degree at most \(n\) if and only if the maps \(T^0,T^1,\ldots,T^n\) are linearly dependent. The authors extend and unify several known results on locally linearly dependent operators. Two applications of their new results, one in algebra and the other in functional analysis, are also given.