Finding matrix linear transformation Question:
Find the $3 \times 3$ matrix $ A$, associated with the linear transformation that projects vectors in $\mathbb{R^3}$ (orthogonally) onto the plane $x+y+z=0$.
I was given this question just as a review question for my class.  I took linear algebra over 2 years ago, so my memory is really fuzzy.
I was given the hints to find the matrix $A$ by thinking of it as a composition of a rotation, simpler projection, then another rotation.  
Thoughts:
I know the normal vector of the plane is $<1,1,1>$. I think this may be a useful fact.  Also after googling a bit I found that the matrix .  Perhaps, I could work with this, although I honestly don't really know exactly how to proceed. I know there are many projections onto this plane, but I don't know exactly how to find an orthogonal one. Any help would be much appreciated.

All you have to do is to find where the transformation sends the
basic unit vectors
$e_1=\pmatrix{1\\0\\0}$, $e_2$ and $e_3$ to, since these will be
the columns of your transformation matrix.
A normal vector to the plane $x+y+z=0$ is $n=\pmatrix{1\\1\\1}$
so $e_1$ will be sent to $e_1+\alpha n=\pmatrix{1+\alpha\\\alpha\\\alpha}$ for some $\alpha$. What then should $\alpha$ be?
