Number 31 and 33 please in R", With k >n.dentUse a theorem from Chapter 1 to explain why S cannot be abasis for R".vectoSpanExercises 31 and 32 reveal an important connection between lin-ear independence and linear transformations and provide practiceusing the definition of linear dependence. Let V and W bevector spaces, let T: V W be a linear transformation, and let{V1,..., Vp} be a subset of V.36. (M] Lwhere= In31. Show that if {V1,...,V,} is linearly dependent in V, thenthe set of images, {T(v),...,T(v,)}, is linearly depen-dent in W. This fact shows that if a linear transforma-V1 =tion maps a set {V1,...,Vp} onto a linearly independent set{T(v)), ... T(vp}}, then the original set is linearly indepen-dent, too (because it cannot be linearly dependent).Find bain Secti32. Suppose that T is a one-to-one transformation, so that anequation T(u) = T(v) always implies u v. Show that ifthe set of images {T(v),..., T(v,)} is linearly dependent,then {v,..., vp} is linearly dependent. This fact shows thata one-to-one linear transformation maps a linearly indepen-dent set onto a linearly independent set (because in this casethe set of images cannot be linearly dependent).37. [M] ShpendentC1 t +Equatiospecificof enou33. Consider the polynomials p, (t) = 1+12 and p,(t) = 1-t2. Is {P1, P2} a linearly independent set in P3? Why or whynot?38. [M] ShpendentЕxercis34. Consider the polynomials p, (1) = 1+t, p2(t)=1-t, and= 2 (for all t). By inspection, write a linear depen-SectionWEBSOLUTIONS TO PRACTICE PROBE1. Let A = [V1 V2]. Row operations1-2A =3

Name: Number 31 and 33 please in R", With k >n.dentUse a theorem from Chapte
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Description: Number 31 and 33 please in R", With k >n.dentUse a theorem from Chapter 1 to explain why S cannot be abasis for R".vectoSpanExercises 31 and 32 reveal an important connection between lin-ear independence and linear transformations and provide practic