Linear Algebra

Section II. Linear Independence 1071.35 In R 4 , what is the biggest linearly independent set you can find? The smallest?The biggest linearly dependent set? The smallest? (‘Biggest’ and ‘smallest’ meanthat there are no supersets or subsets with the same property.)̌ 1.36 Linear independence and linear dependence are properties of sets. We can thusnaturally ask how the properties of linear independence and dependence act withrespect to the familiar elementary set relations and operations. In this body of thissubsection we have covered the subset and superset relations. We can also considerthe operations of intersection, complementation, and union.(a) How does linear independence relate to intersection: can an intersection oflinearly 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 linearlyindependent.1.37 Continued from prior exercise. What is the interaction between the propertyof linear independence and the operation of union?(a) We might conjecture that the union S∪T of linearly independent sets is linearlyindependent if and only if their spans have a trivial intersection [S] ∩ [T] = {⃗0}.What is wrong with this argument for the ‘if’ direction of that conjecture? “Ifthe union S ∪ T is linearly independent then the only solution to c 1 ⃗s 1 + · · · +c n ⃗s n + d 1⃗t 1 + · · · + d m⃗t m = ⃗0 is the trivial one c 1 = 0, . . . , d m = 0. So anymember of the intersection of the spans must be the zero vector because inc 1 ⃗s 1 + · · · + c n ⃗s n = d 1⃗t 1 + · · · + d m⃗t m each scalar is zero.”(b) Give an example showing that the conjecture is false.(c) Find linearly independent sets S and T so that the union of S − (S ∩ T) andT −(S∩T) is linearly independent, but the union S∪T is not linearly independent.(d) Characterize when the union of two linearly independent sets is linearlyindependent, in terms of the intersection of spans.̌ 1.38 For Corollary 1.14,(a) fill in the induction for the proof;(b) give an alternate proof that starts with the empty set and builds a sequenceof linearly independent subsets of the given finite set until one appears with thesame span as the given set.1.39 With a some calculation we can get formulas to determine whether or not a setof vectors is linearly independent.(a) Show that this subset of R 2 ( ( a b{ , }c)d)is linearly independent if and only if ad − bc ≠ 0.(b) Show that this subset of R 3 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞a b c{ ⎝d⎠ , ⎝e⎠ , ⎝f⎠}g h iis linearly independent iff aei + bfg + cdh − hfa − idb − gec ≠ 0.(c) When is this subset of R 3 ⎛ ⎞ ⎛ ⎞a b{ ⎝d⎠ , ⎝e⎠}g h