Standard matrix of an orthogonal projection Okay so I have a vector $\textbf{v}=\begin{bmatrix}1\\-1\\1\end{bmatrix} $. I want to calculate the orthogonal projection of the vector $\textbf{u}=\begin{bmatrix}1\\0\\0\end{bmatrix}$ onto $\textbf{v}$, find the standard matrix of it, give a geometrical argument for whether it is injective/surjective and give a geometrical argument to determine the kernel, null space, image and column space of that standard matrix.
So what I did first was that I calculated the projection using that 
$$P_\textbf{v}(\textbf{u}) = \frac{\langle \textbf{v},\textbf{u} \rangle}{\langle\textbf{v},\textbf{v} \rangle}\textbf{v}=\frac{[1,-1,1]\cdot[1,0,0]}{[1,-1,1]\cdot[1,-1,1]}[1,-1,1]=[\frac{1}{3},-\frac{1}{3},\frac{1}{3}]$$
Which is simple enough, but I don't really know how to use this to create a standard matrix. Also I have no idea how to reason geometrically for the questions asked further, so if someone could push me in the right direction, that would be great!
 A: For any linear map $\varphi:\Bbb R^n\to \Bbb R^m$, its standard matrix $M$ can be obtained by putting $\varphi(e_i)$ in the $i$th column, where $(e_1,\dots, e_n)$ is the standard basis of $\Bbb R^n$.
Note that this also means that $\varphi(v)=M\cdot v$ holds for each basis vector, and hence, by taking linear combinations, it must hold for all vectors $v\in\Bbb R^n$. 
In the case of the orthogonal projection to a vector $v$, we will obtain $P_v=Q:=\frac{vv^T}{\|v\|^2}$, since it satisfies $Qv=v$ and $Qw=0$ if $w\perp v$.
For the geometrical arguments, draw what $P_v$ being an orthogonal projection means. 
Its range (=image=column space) is just the line of $v$ (consisting only of scalar multiples of $v$), which is one dimensional, so the rank is $1$, and it's not surjective. 
Its kernel (=null space) consists of exactly the vectors perpendicular to $v$. In an $n$ dimensional space, the orthogonal subspace of a nonzero vector is $n-1$ dimensional, now it's $2$, and it also means that it's not injective, as there are (a whole plane of) nonzero vectors that are mapped to $0$.
