Subsection 2.1.2 Linear Dependence and Independence
Two non - zero vectors u and v of a vector space are said to be linearly dependent if one is a scalar multiple of the other i.e. if \(u = c v\) where \(c\) is any scalar. In other words, \(u\) and \(v\) are linearly dependent if
\begin{equation}
au+bv =0\tag{2.1.1}
\end{equation}
\([\,\because au=-bv,\hspace{3pt} \text{or,} \hspace{3pt} u=(-b/a)v =cv ]\,\)
In eqn. (2.1.1), at least one scalar does not equal to zero. The concepts of linear dependence and independence can also be extended to more than two vectors. A set of \(n\) vectors \(\left\{u_{i}\right\}\) is said to be linearly dependent if there exists a corresponding set of scalars \(\left\{\alpha_{i}\right\}\text{,}\) not all zero, such that
\begin{equation}
\sum\limits_{i=1}^{n}\alpha_{i}u_{i}=0\tag{2.1.2}
\end{equation}
If \(\sum\limits_{i=1}^{n}\beta_{i}v_{i}=0\) and the set of scalars \(\beta =0\) for all \(i\text{,}\) then the set of vectors \(\left\{v_{i}\right\}\) is said to be linearly independent.
\begin{equation}
\Rightarrow \lambda=-\frac{(v_{1},u_{2})}{(v_{1},v_{1})}\tag{2.1.3}
\end{equation}
Thus we have two orthogonal vectors \(v_{1}\) and \(v_{2}\text{.}\)
