1. Let H = {f(X) ∈ R[X] | f(0) = 0}. a. Show that H = X · R[X] = {X · f(X)| f(X) ∈ R[X]} = Span{X, X2 , X3 , . . .}. b. Use the preceding part to show that H is a subspace of R[X]. Find a basis for H. c. What can you say about Ha = {f(X) ∈ R[X] | f(a) = 0}, where a ∈ R is arbitrary?
2.. Let V denote the vector space of all sequences {an} ∞ n=1 of real numbers. We will say that {an} is eventually zero if there is an N ∈ N so that an = 0 for all n ≥ N (N can vary with the sequence). That is, {an} = {a1, a2, . . . , aN−1, 0, 0, 0, . . .}. Let H denote the set of all eventually zero sequences. Show that H is a subspace of V . Find a basis of H.
3. Let a, b ∈ R and set H = {e ax(p(x) cos bx + q(x) sin bx)| p(x), q(x) ∈ R[x]} . Show that H is a subspace of C 1 (R). Show that this conclusion remains valid if we replace R[x] in the definition of H with Pn. Find a basis for H in both cases. [Remark. You’ll need to know that the function tan x is not a rational function of x. Why is this true? Think in terms of vertical asymptotes.]
4. Show that the set (field) of complex numbers C is an R-vector space. Find an R-basis for C.
2.. Let V denote the vector space of all sequences {an} ∞ n=1 of real numbers. We will say that {an} is eventually zero if there is an N ∈ N so that an = 0 for all n ≥ N (N can vary with the sequence). That is, {an} = {a1, a2, . . . , aN−1, 0, 0, 0, . . .}. Let H denote the set of all eventually zero sequences. Show that H is a subspace of V . Find a basis of H.
3. Let a, b ∈ R and set H = {e ax(p(x) cos bx + q(x) sin bx)| p(x), q(x) ∈ R[x]} . Show that H is a subspace of C 1 (R). Show that this conclusion remains valid if we replace R[x] in the definition of H with Pn. Find a basis for H in both cases. [Remark. You’ll need to know that the function tan x is not a rational function of x. Why is this true? Think in terms of vertical asymptotes.]
4. Show that the set (field) of complex numbers C is an R-vector space. Find an R-basis for C.