The Gram-Schmidt process is a constructive algorithm in linear algebra that transforms any basis for a subspace into an orthogonal or orthonormal basis by iteratively subtracting the projection of subsequent vectors onto previously constructed orthogonal components. This mechanism underpins QR decomposition, which factorizes an invertible square matrix $A$ uniquely (up to scaling) as the product of an orthogonal matrix $Q$ and an upper triangular matrix $R$. The theory relies on formal definitions regarding vector projections, orthogonality conditions ($v_i^T v_j = 0$), and linear independence within $\mathbb{R}^n$, serving as a foundational technique for solving least squares problems and analyzing the spectral properties of matrices.
The Gram-Schmidt process is a constructive algorithm in linear algebra that transforms any basis for a subspace into an orthogonal or orthonormal basis by iteratively subtracting the projection of su…