Question-93
Let \(\displaystyle S=\{v_{1} ,\cdots ,v_{n}\}\) be a finite set of vectors in a vector space. Is the following statement true or false?
\(\displaystyle S\) is linearly dependent if and only if \(\displaystyle v_{i}\) is in the span of \(\displaystyle S-\{v_{i}\}\) for every \(\displaystyle v_{i} \in S\).
This is false. It is sufficient if there is at least one vector in \(\displaystyle S\) that is in the span of the remaining. The condition that every vector in \(\displaystyle S\) should be in the span of the remaining vectors is too hard a constraint for linear dependence.
As a counter-example, consider:
\[ \begin{equation*} S=\{( 1,0,0,0) ,( 0,1,0,0) ,( 1,1,0,0) ,( 0,0,0,1)\} \end{equation*} \]
\(\displaystyle S\) is linearly dependent, but \(\displaystyle ( 0,0,0,1)\) is not in the span of the first three vectors.
The proper definition is:
A finite set of vectors \(S = \{v_1, \cdots, v_n\}\) is linearly dependent if there is at least one vector in \(S\) that belongs to the span of the remaining vectors.
This definition of linear dependence continues to hold if the set has only one element. If \(S = \{0\}\), then \(0\) is in the span of \(S - \{0\} = \phi\). Recall that the span of the empty set is the trivial subspace \(\{0\}\).