# Linear Algebra - Linear Independence
## Definition
In any vector space, a set of vector is 'linear independent' if none of its element is a linear combination of others from the set. Otherwise, it is it is linear dependent.
## Lemma
1. If a linear combination of an independent set of vectors equals $\vec{0}$, then all coefficiant is 0.
$$c_1u_1+c_2u_2+...+c_du_d=\vec{0}\rightarrow c_1=c_2=...=c_d=0$$
3. For a linear independent set $\{u_1,u_2,...u_d\}$
$$ker(\left[ {\begin{array}{} |&|&...&|\\u_1 & u_2 & ... & u_d\\|&|&...&| \end{array} } \right])=\vec{0}$$