Linear Algebra - Formal Power Series
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Question 1: Let x be a variable. Define a formal power series in the variable x over a field F to be a sum of the form
a_0 + (a_1)(x) + (a_2)(x^2) + (a_3)(x^3) + ... = SUM (t=0, infinity) (a_t)(x^t)
with a_t is a real number of F. Let F[[x]] be the set of all formal powers series in x over F. Define an addition on F[[x]] by adding the coefficients of like powers of x, and define a scalar multiplication (with scalars in F) by multiplying all coefficients by the scalar. In other words (see attached).
Prove that F[[x]] is a vector space over F with the given addition and scalar multiplication
Question 2: Let P(F) be the set of all polynomials over a field F. For each of the following, determine (with proof) if the subset is a subspace of P(F).
i) (P_m)(F) = {p is an element P(F) | deg(p) <= m}
ii) U = {p is an element of P(F) | deg(p) = 4}
iii) W = {p is an element of P(F) | p(1) = 0}
iv) W = {p is an element of P(F) | p(1) =/ 0}
Question 3: Suppose that U and W are finitely-generated subspaces of a vector space V. Prove that U + W is also finitely-generated by finding an explicit spanning set of U + W.
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Question 1. Let us verify the axioms of a vector space.
The addition is commutative. Indeed,
∑_(i=0)^∞▒〖a_i x^i 〗+∑_(i=0)^∞▒〖b_i x^i 〗=∑_(i=0)^∞▒〖(a_i+b_i 〖)x〗^i 〗=∑_(i=0)^∞▒〖(b_i+a_i 〖)x〗^i 〗=∑_(i=0)^∞▒〖b_i x^i 〗+∑_(i=0)^∞▒〖a_i x^i 〗
(ii ) The addition is associative . Indeed,
(∑_(i=0)^∞▒〖a_i x^i 〗+∑_(i=0)^∞▒〖b_i x^i 〗)+ ∑_(i=0)^∞▒〖c_i x^i 〗=∑_(i=0)^∞▒〖((a_i+b_i 〖)+c_i)x〗^i 〗=∑_(i=0)^∞▒〖(a_i+〖(b〗_i 〖+c_i))x〗^i 〗= ∑_(i=0)^∞▒〖a_i x^i 〗+∑_(i=0)^∞▒〖〖(b〗_i+〖c_i)x〗^i 〗=∑_(i=0)^∞▒〖a_i x^i 〗+(∑_(i=0)^∞▒〖b_i x^i 〗+ ∑_(i=0)^∞▒〖c_i x^i)〗
(iiI) We have
∑_(i=0)^∞▒〖a_i x^i 〗+∑_(i=0)^∞▒〖0x^i 〗=∑_(i=0)^∞▒〖(a_i+0〖)x〗^i 〗=∑_(i=0)^∞▒〖a_i x^i 〗.
Therefore, F[[x]] contains a zero vector.
(iv) We have
∑_(i=0)^∞▒〖a_i x^i 〗+∑_(i=0)^∞▒〖〖(-a〗_i)x^i 〗=∑_(i=0)^∞▒〖0x^i 〗
Therefore, any vector of F[[x]] has an inverse,
(v) We have
1∑_(i=0)^∞▒〖a_i x^i 〗=∑_(i=0)^∞▒〖〖(1a〗_i)x^i 〗=∑_(i=0)^∞▒〖a_i x^i 〗
(vi) For any real numbers r and r', ...
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