# What does it mean for a linear system to be linearly independent?

Oct 22, 2015

Consider a set S of finite dimensional vectors $S = \left\{{v}_{1} , {v}_{2} , \ldots . {v}_{n}\right\} \in {\mathbb{R}}^{n}$

Let ${\alpha}_{1} , {\alpha}_{2} , \ldots . , {\alpha}_{n} \in \mathbb{R}$ be scalars.

Now consider the vector equation

${\alpha}_{1} {v}_{1} + {\alpha}_{2} {v}_{2} + \ldots . . + {\alpha}_{n} {v}_{n} = 0$

If the only solution to this equation is ${\alpha}_{1} = {\alpha}_{2} = \ldots . = {\alpha}_{n} = 0$, then the set Sof vectors is said to be linearly independent.

If however other solutions to this equation exist in addition to the trivial solution where all the scalars are zero, then the set S of vectors is said to be linearly dependant.