Answer:
Yes. It is a vector space over the field of rational numbers
Step-by-step explanation:
An element of the set has the form
where are rational coefficients.
The operations of addition and scalar multiplication are defined as follows:
The properties that , together the operations of vector addition and scalar multiplication, must satisfy are:
- Conmutativity
- Associativity of addition and scalar multiplication
- Additive Identity
- Additive inverse
- Multiplicative Identity
- Distributive properties.
This is not difficult with the definitions given. The most important part is to show that has a additive identity, which is the zero polynomial, that is closed under vector addition and scalar multiplication. This last properties comes from the fact that is a field, then it is closed under sum and multiplication.