Exercise 21

Continued from prior exercise. What is the interaction between the property of linear independence and the operation of union?

• We might conjecture that the union $S\cup T$ of linearly independent sets is linearly independent if and only if their spans have a trivial intersection $[S]\cap[T]=\text{$\{\hskip 1.00006pt\vec{0}\hskip 1.00006pt\}$}$. What is wrong with this argument for the ‘if’ direction of that conjecture? “If the union $S\cup T$ is linearly independent then the only solution to $c_{1}\vec{s}_{1}+\cdots+c_{n}\vec{s}_{n}+d_{1}\vec{t}_{1}+\cdots+d_{m}\vec{t}_{m}=\vec{0}$ is the trivial one $c_{1}=0$, …, $d_{m}=0$. So any member of the intersection of the spans must be the zero vector because in $c_{1}\vec{s}_{1}+\cdots+c_{n}\vec{s}_{n}=d_{1}\vec{t}_{1}+\cdots+d_{m}\vec{t}_{m}$ each scalar is zero.”

• Give an example showing that the conjecture is false.

• Find linearly independent sets $S$ and $T$ so that the union of $S-(S\cap T)$ and $T-(S\cap T)$ is linearly independent, but the union $S\cup T$ is not linearly independent.

• Characterize when the union of two linearly independent sets is linearly independent, in terms of the intersection of spans.