Definition.
A set \(S=\{\overrightarrow{v_1},\overrightarrow{v_2},\ldots,\overrightarrow{v_k}\}\) of vectors of a vector space
\(V\) is linearly independent if the only linear combination of vectors in \(S\) that produces
\(\overrightarrow{0}\) is a trivial linear combination., i.e.,
\[c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+\cdots+c_k\overrightarrow{v_k}=\overrightarrow{0}
\implies c_1=c_2=\cdots=c_k=0.\]
\(S=\{\overrightarrow{v_1},\overrightarrow{v_2},\ldots,\overrightarrow{v_k}\}\) is linearly dependent
if \(S\) is not linearly independent, i.e., there are scalars \(c_1,c_2,\ldots,c_k\), not all zero, such that
\[c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+\cdots+c_k\overrightarrow{v_k}=\overrightarrow{0}.\]
Example.
\(\{\overrightarrow{v}\}\) is linearly independent in \(V\) if and only if
\(\overrightarrow{v}\neq \overrightarrow{0_V}\).
\(\{\overrightarrow{e_1},\overrightarrow{e_2},\ldots,\overrightarrow{e_n}\}\) is a linearly independent set of
vectors in \(\mathbb R^n\).
\(\{\overrightarrow{1},\overrightarrow{t},\overrightarrow{t^2},\ldots,\overrightarrow{t^n}\}\) is
a linearly independent set of vectors in \(P_n\).
\(\{\overrightarrow{e_1},\overrightarrow{e_2},\ldots,\overrightarrow{e_n},\ldots\}\) is a linearly independent set
of vectors in \(\mathbb{R}^{\infty}\) where \(\overrightarrow{e_i}\) is the infinite sequence with \(1\) in the
\(i\)th place and \(0\) elsewhere.
\(B=\{\overrightarrow{E_{i,j}}: 1 \leq i \leq m,1 \leq j \leq n\}\) is a linearly independent set of vectors in
\(M_{m, n}(\mathbb R)\) where \(\overrightarrow{E_{i,j}}\) is the \(m\times n\) matrix with \((i,j)\)-entry 1 and \(0\)
elsewhere.
Consider the following three polynomials in \(P_2\): \(\overrightarrow{p_1}(t)=t+2t^2\),
\(\overrightarrow{p_2}(t)=2+2t^2\), and \(\overrightarrow{p_3}(t)=1-t-t^2\).
Show that \(\{\overrightarrow{p_1},\;\overrightarrow{p_2},\;\overrightarrow{p_3}\}\) is a linearly dependent set
in \(P_2\).
Solution.
Suppose \(c_1\overrightarrow{p_1}+c_2\overrightarrow{p_2}+c_3\overrightarrow{p_3}=\overrightarrow{0}\) for
some scalars \(c_1,c_2,c_3\). Then for all \(t\),
\[\begin{align*}
(c_1\overrightarrow{p_1}+c_2\overrightarrow{p_2}+c_3\overrightarrow{p_3})(t)&=0\\
c_1\overrightarrow{p_1}(t)+c_2\overrightarrow{p_2}(t)+c_3\overrightarrow{p_3}(t)&=0\\
c_1(t+2t^2)+c_2(2+2t^2)+c_3(1-t-t^2)&=0\\
(2c_2+c_3)+(c_1-c_3)t+(2c_1+2c_2-c_3)t^2&=0.
\end{align*}\]
Thus \(2c_2+c_3=0,\;c_1-c_3=0,\;2c_1+2c_2-c_3=0\). One solution is \((c_1,c_2,c_3)=(2,-1,2)\). So \(2\overrightarrow{p_1}-\overrightarrow{p_2}+2\overrightarrow{p_3}=\overrightarrow{0}\)
and \(\{\overrightarrow{p_1},\;\overrightarrow{p_2},\;\overrightarrow{p_3}\}\) is a linearly dependent set in \(P_2\).
Theorem.
A set \(S=\{\overrightarrow{v_1},\overrightarrow{v_2},\ldots,\overrightarrow{v_k}\}\) of \(k\geq 2\) vectors in
a vector space \(V\) is linearly dependent if and only if there exists a vector in \(S\) that is a linear combination
of the other vectors in \(S\).