Let $$\mathbb{C}[[x]] := \{\sum_{n\geq 0} a_n x^n | a_n \in \mathbb{C}\}$$ be the set of formal power series of x and $V$ be the vector space of all series over $\mathbb{C}$.
Let in addition $V_1$ be the set of series for which applies
$$f_a(x) = \sum_{n\geq 0} a_n x^n = \frac{P(x)}{Q(x)}$$ with $Q(x) = 1 + \alpha_1 t +\cdots + \alpha_d t^d$ and a polynomial $P(x)$ having a degree < $d$.
a)
Prove that $V_1$ is a vector subspace of $V$.
b)
Show that the dimension of $V_1$ is $d$ and identify a base of $V_1$.
Hi!
a)
As I know I need to show three things:
- $V_1 \neq \emptyset$
- $\forall x,y \in V_1 : x + y \in V_1$
- $\forall x \in V_1, \forall y \in V : x \cdot y \in V_1$
For (1) I need to show that at least one element exists and therefore $V_1$ is not empty.
If I pick $P(x) = x^2, \; Q(x) = 1 + x + x^2 + x^3$ I get $P(x) /Q(x) = 0$. This by definition is $\in V$, isn't it?
2) How do I add two series?
3) According to Wikipedia the product of two series is given by $$\left( \sum_{n=0}^\infty a_n x^n \right) \left( \sum_{n=0}^\infty b_n x^n \right) = \sum_{n=0}^\infty c_n x^n, \; c_n = \sum_{k=0}^n a_k b_{n-k}$$
I tried to play around with some values, but this all didn't make any sense. How do I show that multiplying an element of $V_1$ with any other series ($\in \mathbb{C}$) is again $\in V_1$?
b)
What is the "dimension" of a formal power series?
Thank you in advance!
Best Answer
Your requirement "3." is wrong. It would actually say that $V_1$ is an ideal of $V$. What you actually need is $c x \in V_1$ for any $c \in {\mathbb C}$ and $x \in V_1$.