Hello everyone, I have a query regarding the Gram-Schmidt factorization: Say I have 3 independent vectors, u, v, w and I used the factorization scheme to get U, V, W vectors that are orthonormal to each other. So U, V, W are orthogonal to each other. Is it also true that V is orthogonal to u (small u) and W is orthogonal to small u and v. In my mind, I am quite convinced it is so. However, what is the exact mathematical reason for this. I think that W is going to be orthogonal to the whole plane described by u and v and V is going to be orthogonal to the line described by u. I am just a tad unsure why this would be. I am trying to understand that when we do QR factorization why we get an upper triangular matrix and that seems to depend on the above statement being true. Many thanks, Luca Edit: I thought about this a bit more and have the following explanation. Please let me know if it sounds plausible If we consider the vectors U and V. They form the basis for a 2D subspace in a higher dimension space. So, the vector W lies in the null space of this 2D subspace and this null space will be orthogonal to the row space of the 2D subspace. Hence, W is orthogonal to u and v as well as they lie in the same subspace. Does this make sense?