:PROPERTIES: :ID: e65e7624-7323-40e2-8b8d-5fe498877e85 :mtime: 20231017011113 20231014015155 :ctime: 20231014015148 :END: #+title: linearly independent #+filetags: :public:project: * Definition Given a list of $n$ vectors \[B = \{\vec{e}_{1} \ldots \vec{e}_{n}\}\] We say that the list of vectors is *linearly independent* if and only if the only solution to the equation \[\lambda_{1}\vec{e}_{1} + \lambda_{2}\vec{e}_{2}+\cdots + \lambda_{n}\vec{e}_{n} = \vec{0}\] is \[\lambda_{1} = 0 , \lambda_{2} = 0 , \ldots , \lambda_{n} = 0\] * Alternate Definition A list of vectors $B$ is said to be *linearly independent* if and only if for all $\vec{r}$, the equation \[\vec{r} = \lambda_{1}\vec{e}_{1} + \cdots +\lambda_{n}\vec{e}_{n}\] has one and only one solution. * Orthogonality If $\vec{e}_{1} , \ldots \vec{e}_{n}$ are all orthogonal and linearly independent, then * How to test for linear independence ** Using [[id:e72ab120-e332-4c9d-bf88-ea02dcd4eb99][scalar triple product]] $\vec{a}, \vec{b}, \vec{c}$ are linearly independent if and only if \[[\vec{a},\vec{b}, \vec{c}] \neq 0\] * Example Problems ** Test if vectors are linearly independent *** Problem Are $\vec{i}, \vec{j}, \vec{i} + \vec{j}$ linearly independent? *** Solution No.