We define linear independence of vectors. We begin with the following.
Definition3.1.1.(Linearly dependent vectors).
Let \(V\) be a vector space over a field \(F\text{.}\) A set of vectors \(\{v_1,v_2,\ldots,v_n\}\subset V\) is said to be linearly dependent over \(F\) or vectors \(v_1,v_2,\ldots,v_n\) are linear dependent if there exists scalars \(\alpha_1,\alpha_2,\ldots,\alpha_n\in F\) not all zero such that
An infinite subset \(S\) of \(V\) is said to be linearly dependent over \(F\) if there is a finite nonempty subset of \(S\) which is linearly dependent over \(F\text{.}\)
Remark3.1.2.
Any subset of \(V\) containing the zero vector is linearly dependent.
If \(v\) and \(w\) are linearly dependent then, there exists a nonzero \(\alpha\) (resp., a nonzero \(\beta\)) such that \(v=\alpha w\) (resp., \(w=\beta v\)). In other words, \(v\) and \(w\) lie on the same line (refer Definition 2.5.15) and hence \(v,w\) are collinear.
The relation ’\(v\) and \(w\) are linearly dependent’ is an equivalence relation on the set of nonzero vectors of \(V\) (Verify this statement). The equivalence classes are lines with zero removed.
Definition3.1.3.(Linearly independent vectors).
A nonempty set \(S\) of vectors in \(V\) are said to be linearly independent over \(F\) if \(S\) is not linearly dependent over \(F\text{.}\)
In other words, \(S\) is linearly independent over \(F\) if for any distinct vectors \(v_1,v_2,\ldots,v_r\in S\)
\begin{equation*}
\alpha_1 v_1+\alpha_2 v_2+\cdots+\alpha_r v_r=0\quad\text{implies that}\quad\alpha_i=0\quad\text{for all }i=1,2,\ldots,r.
\end{equation*}
Remark3.1.4.
A nonempty set \(S\) of vectors is linearly independent if and only if each nonempty finite subset of \(S\) is linearly independent.
Remark3.1.5.
We make the following conventions.
The empty set is linearly independent.
The span of the empty set is the zero space \(\{0\}\text{.}\)
