What are polynomials of degree less than $N$ in $x$?

Question

I don't understand what this question is asking me:

Consider the collection of all polynomials (with complex coefficients) of degree less than $N$ in $x$.

Does this mean the polynomials like $$ Ax^{N-1},\ Bx^{N-2}, \ Cx^{N-3}, \ ...$$ where $A$,$B$,$C$ are complex coefficients? If so, when does this series end and what is its null vector?

Answer 1

The polynomials of degree less than $N$ with complex coefficients are given by $$\left \{\sum_{i=0}^{N-1} a_i x^i | a_i \in \mathbb{C}\right \} $$ The important thing is to note that a polynomial does not have negative powers of $x$.
This is as you have rightly stated a vector space (which can be visualised as $\mathbb{C}^{N}$). The null vector is the only polynomial which can be added to every other polynomial without changing this one - I leave that to you.
It surprises me, though, that you were not able to find this by a simple search on google ...

Answer 2

What it means is that you have the set $S=\{(Ax^{N-1}+Bx^{N-2}+...+F_{n}),\,(A_{1}x^{N-2}+B_{2}x^{N-3}+...+F_{n1})\,,{(polinomyal)},{(polinomyal)},{(polynomial)}....{(F_{nn)}} \}$

The null vector depends on what operation you're using, if you are multiplying this set, the null vector is 1. Let $\vec{v}=A_{1}x^{N-2}+B_{2}x^{N-3}+...+F_{n1}$ and you define multiplication as the linear operation, then $\vec{v}*1=\vec{v}$ and technically $1$ is a polynomial of degree less that N, if $ N\epsilon \mathbf{N} $