I don't understand the Quotient Space well I'm looking examples of problems a their solutions that involved Quotient Spaces and someone commented the following:
... Let be $V=\Bbb R[x]$ and if $U=\{\text{polynomials $p(x)$ such that $p(0)=0$}\}$, then $\dim(V/U)=1$...
Why the dimesion of $\dim(V/U)=1$?
Can someone explain, please.
Thanks

To find the dimension of $\mathbb R[x]/U$, we must find a basis for it. This is actually pretty easy to do in this particular case just by starting with the standard basis for $\mathbb R[x]$:
$$
\{1, x, x^2, x^3, \ldots\}
$$
Of these basis vectors, nearly all of them are in $U$. For instance, $0^2 = 0$, and therefore $x^2 \in U$ by definition of $U$. So, other than $1$, all of these basis vectors become zero in the quotient $\mathbb R[x]/U$. As a consequence, $\overline 1$ (the image of $1$ in the quotient space) spans all of $\mathbb R[x]/U$. So $\dim(\mathbb R[x]/U)$ is at most $1$.
The possibilities, then, are that $\dim(\mathbb R[x]/U)$ is $0$ or $1$. It is $0$ precisely when $\mathbb R[x]/U = \{\overline 0\}$. This is equivalent to $\mathbb R[x] = U$. But of course this is not true! For instance, $1 \notin U$, so $\mathbb R[x] \neq U$. So we conclude that the dimension is exactly $1$.
