Linear independence and linear dependence are properties of sets. We can thus naturally ask how the properties of linear independence and dependence act with respect to the familiar elementary set relations and operations. In this body of this subsection we have covered the subset and superset relations. We can also consider the operations of intersection, complementation, and union.
(a) How does linear independence relate to intersection: can an intersection of linearly independent sets be independent? Must it be?
(b) How does linear independence relate to complementation?
(c) Show that the union of two linearly independent sets can be linearly independent.
(d) Show that the union of two linearly independent sets need not be linearly independent.