1. ## Orthogonal Prjection Question

Find the matrix for the orthogonal projection onto the subspace of $R^3$ defined
by $3x_1 -x_2 + 3x_3$

2. Originally Posted by flaming
Find the matrix for the orthogonal projection onto the subspace of $R^3$ defined
by $3x_1 -x_2 + 3x_3 \color{red}{} = 0$.
The projection P onto the 1-dimensional subspace spanned by the vector $\mathbf{v} = (3,-1,3)$ is given by $P\mathbf{x} = \langle\mathbf{x},\mathbf{v}\rangle/\|\mathbf{v}\|^2 = \tfrac1{19}(3x_1-x_2+x_3)\mathbf{v}$, where x is the vector (x_1,x_2,x_3). In matrix notation, this looks like $P\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} = \frac1{19}\begin{bmatrix}9x_1-3x_2+9x_3 \\-3x_1+x_2-3x_3 \\9x_1-3x_2+9x_3 \end{bmatrix} = \frac1{19}\begin{bmatrix}9&-3&9 \\-3&1&-3 \\9&-3&9 \end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmat rix}$. So the matrix of P is $M_P = \frac1{19}\begin{bmatrix}9&-3&9 \\-3&1&-3 \\9&-3&9 \end{bmatrix}$.

The 2-dimensional subspace given by $3x_1 -x_2 + 3x_3 = 0$ is the orthogonal complement of the above 1-dimensional subspace. So the projection onto it will have matrix $I-M_P$.