Define:
A set of vectors \(\boldsymbol{u_1, u_2,..., u_n}\) are linearly independent when
\[a_1\boldsymbol{u_1} + a_2\boldsymbol{u_2} + ... + a_n\boldsymbol{u_1} = \boldsymbol{0} \Longleftrightarrow a_1 = a_2 = ... = a_n = 0\]
Define:
A set of vectors \(\boldsymbol{u_1, u_2,..., u_n}\) are linearly dependent if they are not linearly independent.
Remark:
\[\boldsymbol{u_1} = -(a_2\boldsymbol{u_2} + ... a_n\boldsymbol{u_n})/a_1\]Linear dependence implies that \(\boldsymbol{u_i}\) can be expressed as a linear combination of other vectors, e.g.
\(a_1\boldsymbol{u_1} + a_2\boldsymbol{u_2} + ... + a_n\boldsymbol{u_1} = \boldsymbol{0}\) can be expressed as:
\[\begin{bmatrix}\boldsymbol{u_1} & ... & \boldsymbol{u_n}\end{bmatrix} \begin{bmatrix}a_1\\ ...\\ a_n\end{bmatrix} = \boldsymbol{0}\]Let \(U = \begin{bmatrix}\boldsymbol{u_1} & ... & \boldsymbol{u_n}\end{bmatrix}\).
The vectors in \(U\) are linearly independent if only if \(N(U)=\{\boldsymbol{0}\}\).