Let $c_0=\{ (x_n) : x_n \in \mathbb R, x_n \to 0 \}$, let $M=\{(x_n) \in c_0 : x_0+x_2+ \cdots +x_{10}=0 \}$ Then dimension of $c_0/M$ is ? Answer is given to be 1.
My attempt: 9 "coordinates" are needed to specify $M$, so $\dim M=9$.
$\dim(c_0/M)=\dim c_0-\dim M$.
How do I find dim $c_0$?
Alas, as $c_0$ and $M$ are both infinite-dimensional, your formula $\dim(c_0/M)=\dim c_0-\dim M$ doesn't work.
The quotient space is $1$-dimensional. This follows from the First Isomorphism Theorem, applied to the map $T:c_0\to\Bbb R$ given by $T:(x_n)\mapsto x_0+x_2+\cdots+x_{10}$. This is a linear map, and has image $\Bbb R$. By the First Isomorphism Theorem, $\Bbb R$ is isomorphic to $c_0/\ker T$. Of course the kernel of $T$ is $M$.