What does the notation $\mathbb R[x]$ mean?

Question

What does the notation $\mathbb R[x]$ mean?

I thought it was just the set $\mathbb R^n$ but then I read somewhere that my lecturer wrote $\mathbb R[x] = ${$\alpha_0 + \alpha_1x + \alpha_2x^2 + ... + \alpha_nx^n : \alpha_0, ..., \alpha_n \in \mathbb R$}

Edit: The reason why I asked this question was because I had a tutorial question that said:

Check whether a system {$v_1,...,v_m$} of vectors in $\mathbb R^n$ (in $\mathbb R[x]$) is linearly independent.

I just assumed it meant the same thing when they put it in brackets like that. Since it isn't the case, how must I interpret this question.

Answer 1

$\mathbb{R}[x]$ denotes the set of all polynomials with coefficients in $\mathbb{R}$. In particular, this set forms a ring under polynomial addition and multiplication. There is no restriction on the degrees of these polynomials, however, as your post suggests. As GitGud stated in the comments, you need an $n \in \mathbb{N}$ somewhere after the colon in your set builder notation. In particular, note the difference between: $$\{a_0 + a_1x + a_2x^2 + ... + a_nx^n \ | \ a_0, ..., a_n \in \mathbb{R} \}$$ and $$\{a_0 + a_1x + a_2x^2 + ... + a_nx^n \ | \ a_0, ..., a_n \in \mathbb{R} \ \wedge \ n \in \mathbb{N} \}$$

In general, $R[x]$ denotes the set of all polynomials with coefficients in a ring $R$. Common examples include $\mathbb{Q}[x]$ (rational coefficients) and $\mathbb{Z}[x]$ (integer coefficients).