find $M^{\perp}$ in Hilbert space Given $n \in \mathbb{Z}^+$ and $M = \{ (x_1,x_2,..,x_n,0,0,...) \mid x_1,x_2,..,x_n \in \mathbb{R} \}$. Find $M^{\perp}$ in $l^2$?
I can show $M$ is a closed subspace of $l^2$ and a Hilbert space. 
Let $P_M : l^2 \to M$: 
$$x= (x_1,x_2,..,x_n,..) \mapsto (x_1,x_2,..,x_n,0,0,..)$$
$P_M$ is a projection from $l^2$ onto $M$. 
But how can I find $M^{\perp}$, $\mathrm{Im} P_M$ and $\mathrm{Ker} P_M$ ?

Let $X$ be a Hilbert space and $P_H$ be the projection onto a subspace $H$ of $X$. We always have $X = P_HX\oplus (P_HX)^{\perp}$. Recall that the direct sum of two subspaces is the space of all sums of elements from the two spaces assuming they only have zero in common. Specifically, if $X = \ell^2$ and $H$ is a finite projection of $x \in \ell^2$ to its first $n$ coordinates, call this $x^{(n)}$, it follows that $y \in (P_H\ell^2)^{\perp}$ if and only if
$$
y = x - x^{(n)} = (0,0,\dots,0,x_{n+1},x_{n+1}, \dots).
$$
