Prove that a Polynomial ring is a vector space so I shall prove that a Polynomial ring [K] is a vector space. How do I do that?
I was thinking of just going down all axioms one by one.. but I don't really know how to prove them for a polynomial ring.
 A: To check that $K[X]$ is a vector space, you need to know how addition and multiplication by elements of $K$ are defined in $K[X]$ (the vector space structure ignores the more general multiplication of polynomials with each other). Polynomials are completely determined by knowing for each $i$ their coefficient of $X^i$. Addition is defined so that the coefficient of $X^i$ in $P+Q$ is just the sum of the coefficients of $X^i$ in $P$ and in $Q$, for any $i$. Multiplication by a scalar $\lambda\in K$ also does not mix coefficients from different powers of $X$: the coefficient of $X^i$ in $\lambda P$ is $\lambda$ times the coefficient of $X^i$ in$~P$. This means that to check the axioms of vector spaces, all of which state that certain equalities always hold, you can fix some $i$ (symbolically: you never actually choose a concrete value for it), and check that the coefficients of $X^i$ on both sides of the equality match. This is very easy, for each axiom.
A: Given any set $A$, the set of all functions $A \to K$, with  pointwise addition and scalar multiplication, is a vector space over $K$.
The set of all functions $A \to K$ that have finite support is a vector subspace.
The ring of polynomials $K[X]$ can be identified with the set of all functions $\mathbb N \to K$ that have finite support and so is a vector space over $K$.
