A linear transformation $T$ is a projection if $T^2 = T$, and $T$ is a reflection if $T^2 = I$.
I have $T(x,y) = (-x -2y, y)$. I have already shown that $T$ is a reflection, but I also need to determine if $T$ is orthogonal, and I'm not sure how.
I have $T(x,y) = (x+y,0)$. I know it's a projection but, again, I need to determine if $T$ is orthogonal and I'm not sure how.
My intuition was to dot product $T(x,y)$ with $T(a,b)$ and verify if it's equal to $0$.
Guide:
A transformation is orthogonal if $\langle u, v\rangle = \langle Tu, Tv\rangle$.
In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of $V$. The columns of the matrix form another orthonormal basis of $V$.
$$T(x,y) = \begin{bmatrix} -1 & -2 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}=Ax$$
You just have to verify if $A^TA=I$, if it is, then it is orthogonal.