Projection on the subspace perpendicular to a vector

Question

I am running an algorithm and during one of the steps of the algorithm, I have to update a matrix $B$ but projecting every column of the matrix, $B_j$, j=1,...,k, on the subspace perpendicular to a specific column, $B_i$. How do I compute this projection? How can a compute a projection of a vector on a subspace orthogonal to a given vector $B_i$? A similar problem consists of finding an orthonormal basis for the subspace of $B$ orthogonal to a vector $B_i$. How can I compute this? Any help would be appreciated... I'd like to compute it with matrices

Answer 1

Let $v$ be a non-zero vector, then the orthogonal projection on the subspace orthogonal to $v$ is given by $$P(w) = w - \frac{v\cdot w}{\|v\|^2}v,$$ where $v\cdot w$ is the usual scalar product.