The Gram-Schmidt process is a constructive algorithm in linear algebra that transforms any given basis into an orthogonal or orthonormal basis for the same subspace, thereby ensuring that every vector in the span can be uniquely decomposed into components along these new axes. This method relies on the formal projection theorem, which dictates subtracting sequential projections of subsequent vectors onto previously constructed orthogonal vectors to eliminate redundant directional information. Consequently, it provides a mechanism for generating mutually perpendicular basis vectors from an arbitrary set without altering the underlying geometric subspace defined by the original input vectors.
The Gram-Schmidt process is a constructive algorithm in linear algebra that transforms any given basis into an orthogonal or orthonormal basis for the same subspace, thereby ensuring that every vecto…