2. Let F be the vector space of continuous functions f : R. R and let H be the subspace of F spanned by {1, sin(t), cos(t), cos(20). (a) (Spt) Let K be the set of functions K = sin2(t) + boos2(t) : a, b E R}. Determine whether K is a vector subspace of H or not. (Hint: some standard trigonometric identities could be useful.) Justify your answer. (b) (Spt) Let L be the set of functions L = {sin(t) + a cos(2t) : a E R}. Determine whether L is a vector subspace of H or not. Justify your answer. (c) (Spt) Consider R as a real vector space (under the usual real number addition and multiplica-tion). Let T : H —> R be the function given by T(f) = fir fit) di. Show that T is a linear transformation. (d) (Spt) Find a spanning set for ker(T). Justify your answer. (e) (Spt) Let D : H —> F be given by D(f) = , the first derivative. Show that D is a linear transformation. (f) (Spt) Find a spanning set for ker(D), where D is the linear transformation from part (e). Justify your answer.

-6 is the correct option from the above given options

step-by-step explanation:

i found the life tables for the year 2012. 

life table of males at age 30. probability of dying between ages x and x+1

total population:  

29–30 . . . . . . . . . . . . 0.001030

30–31 . . . . . . . . . . . . 0.001061

males:  

29–30 . . . . . . . . . . . . 0.001420

30–31 . . . . . . . . . . . . 0.001448

white population

29–30 . . . . . . . . . . . . 0.001005

30–31 . . . . . . . . . . . . 0.001037

white males

29–30 . . . . . . . . . . . . 0.001372

30–31 . . . . . . . . . . . . 0.001399

non hispanic white population

29–30 . . . . . . . . . . . . 0.001083

30–31 . . . . . . . . . . . . 0.001116

non hispanic white males

29–30 . . . . . . . . . . . . 0.001477

30–31 . . . . . . . . . . . . 0.001503

jacob, non hispanic white male. 30 years old.

probability that jacob lives to be 30 years old. 

since the given probabilities are probabilities of dying, i think we simply subtract it from 1 to get the probability of living. we will use the probability of non hispanic white male.

                                          probability of living to be 30 y.o.

29 - 30 : 1 - 0.001477 = 0.998523

30 - 31 : 1 - 0.001503 = 0.998497