[Math] Orthogonal Projection of matrix onto subspace

Question

Let's say I have the subspace

$$S=\{(X_1,X_2,X_3,X_4)\mid~~6X_1 – 2X_2 + 4X_3-10X_4 = 0\}$$

How do I go about finding the matrix which is the orthogonal projection onto this subspace?

Answer 1

The relation defining your space is $$ X \in S \quad \Leftrightarrow \quad \langle X, (6, -2, 4, -10) \rangle = 0 $$ where $\langle \cdot, \cdot \rangle$ is the dot product. So one very obvious guess of a vector that is orthogonal to all $X$ in $S$ is $(6, -2, 4, -10)$. The orthogonal complement of $S$ is, therefore, the space generated by $u = (6, -2, 4, -10)$. (By dimension counting, you know that $1$ generator is enough.) The projection operation is $$ P(X) = X - \frac{\langle X, u\rangle}{\langle u, u\rangle}u = X - \frac{uu^T}{u^Tu}X = \left(I - \frac{uu^T}{u^Tu}\right)X. $$