Consider the following subspaces of $\in R^{3}$.
$U: x_{1}+2x_{2}+3x_{3}=0$
$W= span$ {$(1, -1, 1)$}
Show that $\in R^{3}= W \bigoplus U$ and write the expression in coordinates of the projection on $W // U$
After deducing the span of the subspace $U$ I've verified the linear independency by
$a(1,-1,1) + b(-2,1,0) + c(-3, 0, 1)= (0,0,0)$ $\leftrightarrow a=b=c=0$
Then, I concluded that $W \cap U=0$
So, consequently for the dimension theorem we have that $ R^{3}= W \bigoplus U$
Now, I'm having problems with the second part of the exercise.
I would be very thankful if someone can explain me geometrically what does it mean a projection on $W$ parallel to $U$
Geometrically we start out from a point $v\in\Bbb R^3$ and move it parallelly with $U$ to arrive to a point $P(v)$ in $W$.
Put in other words, $P(v)$ is the intersection of $W$ and the affine subspace (plane in this case) $U+v$, which is a unique point if $\Bbb R^3=U\oplus W$.
Algebraically we can simply define $P(v)=P(u+w)=w$ where $u\in U,\,w\in W$ are the unique elements satisfying $v=u+w$.
Now either you directly calculate $P(e_i)$ for the standard basis vectors $e_1,e_2,e_3$, or write up the matrix of $P$ in the basis you found and apply basis transformation.