Is $M_n(\mathbb{R})[x]$ is a vector space? If I consider set of all polynomials $M_n(\mathbb{R})[x]$ i.e set of objects of the form $A_0+xA_1+x^2A_2+\dots$ are they form vector space over $\mathbb{R}$ or $\mathbb{C}$?
Thank you for help.
 A: The answer depends on the definitions of the addition and scalar multiplication in the vector space.
If we use the ''natural'' definitions:
$$
(A_0+xA_1+\cdots +x^nA_n)+(B_0+xB_1+\cdots +x^nB_n)=(A_0+B_0)+x(A_1+B_1)+\cdots +x^n(A_n+B_n)
$$
and
$$
\alpha(A_0+xA_1+\cdots +x^nA_n)=(\alpha A_0)+x(\alpha A_1)+\cdots + x^n(\alpha A_n)
$$
than $\alpha $ must be an element of $\mathbb{R}$ if we want that  $\alpha A_i \in M_n(\mathbb{R})$.
In this case it is simple to show that the axioms of a vector space, over $\mathbb{R}$, are verified as a consequence of the fact that $M_n(\mathbb{R})$ is a vector space over $\mathbb{R}$.
A: You just have to prove that your set fulfills all the axioms needed to form a  vector space. You can find those axioms in the next page: https://en.wikipedia.org/wiki/Vector_space
To the answer of your question, if you do proof that the axioms hold, you will see that your set forms a vectorial space.
A: As Vincent said: this is nothing but definition-checking (https://en.m.wikipedia.org/wiki/Vector_space)
Just a side note: polynomials are *finite expressions ** of the form
$$A_0+A_1x+\cdots+A_nx^n,$$
i.e. I'd recommed not to put the dots without the closing $A_nx^n$.
