Let G: $\displaystyle R^2 -> R^2$ be orthogonal projection onto the line y = -x, show that G has a standard matrix of $\displaystyle \left[\begin{array}{cc}0.5&-0.5\\-0.5&0.5\end{array}\right] $
The simplest way to find the matrix corresponding to a given linear transformation in a given basis is to apply the linear transformation to each of the basis vectors in turn, writing the result in terms of the basis. The coefficients of each linear combination are the numbers in that column of the matrix.
The "standard" basis for $\displaystyle R^2$ is {<1, 0>, <0, 1>}. What is the projection of <1, 0> on y= -x (with direction vector <1, -1>)? What is the projection of <0, 1> on that line?
Warning: the matrix is NOT $\displaystyle
\left[\begin{array}{cc}0.5&-0.5\\-0.5&0.5\end{array}\right]
$
You must have copied something wrong.