Question

Give an example of a vector space V finitely generated over a field F , together with nonempty subsets B1, B2, and B3 of V satisfying the following conditions: (1) Each Bi is linearly independent;
(2) For each 1≤I ̸= j ≤3 there exists a basis of V containing Bi ∪Bj;

(3) There is no basis of V containing B1 ∪ B2 ∪ B3.

Answer 1

on hat r=124 be a veeter Space over field IR. and set {4, ez, ez, eus be a standard basis of 1R4. And also Bete}, B2={18) Then clearly B. , B₂ and Bg are linearly independent, By es singleton Hence linearly melependent B2 is ergens auto set of two lineally indenelery veetera 2) L4 B, U B2 = of G, ez, eus Hence I a bases reges, ezilas of 1R4 wheel conteyn. BU B2 Now B₂U B z = TO) 5 Now Gues = f ($), (1.0), ( BUBB = { (!).(6), (8) (8 wheeh also itself Wheel itself is a basis of 1R4 BLUB 3) out oue, we = f(509) 0: bass 9 1R9 101 1 1 1 cardenilety 7 B, UB₂U B3 is 5 but is 4 Hence there does not exust Confarnung BLUB Z OB3 dimension A IR any bases