Gram-Schmidt Orthogonalization produces a an orthogonal basis for a set of linearly independent vectors. So, the idea of getting a vector’s corresponding vector in the orthogonal basis is substracting the vector’s projections onto the preceding basis vectors by the vector itself. Then whatever left is orthogonal to all the preceding basis vectors.

My question was, why that further substractions of the projections do not affect the orthogonalization with previous basis vectors??? After a while of thinking.. I realize that it’s because further substractions all happen in spaces orthogonal to previous basis vectors, so of course whatever linear operations of vectors on the orthogonal spaces do not affect the resulting vector’s orthogonalization with previous basis vectors..