Our previous definition of linear independence (Definition LICV) employed a relation of linear dependence that was a linear combination on one side of an equality and a zero vector on the other side. As a linear combination in a vector space (Definition LC) depends only on vector addition and scalar multiplication, and every vector space must have a zero vector (Property Z), we can extend our definition of linear independence from the setting of $\complex{m}$ to the setting of a general vector space $V$ with almost no changes. Compare these next two definitions with Definition RLDCV and Definition LICV.
Suppose that $V$ is a vector space. Given a set of vectors $S=\set{\vectorlist{u}{n}}$, an equation of the form \begin{equation*} \lincombo{\alpha}{u}{n}=\zerovector \end{equation*} is a relation of linear dependence on $S$. If this equation is formed in a trivial fashion, i.e. $\alpha_i=0$, $1\leq i\leq n$, then we say it is a trivial relation of linear dependence on $S$.
Suppose that $V$ is a vector space. The set of vectors $S=\set{\vectorlist{u}{n}}$ from $V$ is linearly dependent if there is a relation of linear dependence on $S$ that is not trivial. In the case where the only relation of linear dependence on $S$ is the trivial one, then $S$ is a linearly independent set of vectors.
Notice the emphasis on the word “only.” This might remind you of the definition of a nonsingular matrix, where if the matrix is employed as the coefficient matrix of a homogeneous system then the only solution is the trivial one.