How to find a matrix of a linear transformation?

Question

Let $V = \text{span}\{(1, 1, 1),(-1, 1, 2)\}$, and let $T:\Bbb R^3 \mapsto\Bbb R^3$ be the linear transformation given by the orthogonal projection onto $V$. What is the standard matrix of $T$? Please explain.

Answer 1

Let $u=(x,y,z)\in V^\perp$ then we have using the inner product:

$$x+y+z=-x+y+2z=0$$ so by letting $z=2$ we find $y=-3$ and $x=1$ hence we normalize and we have $$u=\frac1{\sqrt{14}}(1,-3,2)$$ is a unit vector and $V^\perp=\langle u\rangle$. Now the projection matrix on to $V^\perp$ is $$P_{V^\perp}=\frac{1}{14}\left(\begin{matrix}1&-3&2\\-3&9&-6\\2&-6&4\end{matrix}\right)$$ and finally since $$P_V+P_{V^\perp}=I_3$$ the result follows.