The least squares method provides a mechanism to solve inconsistent linear systems $Ax = b$ by minimizing the Euclidean norm $\|Ax - b\|$ between two vectors, which corresponds to finding the orthogonal projection of $b$ onto the column space of matrix $A$. This concept relies on fundamental properties from linear algebra and functional analysis regarding norms in Hilbert spaces, specifically utilizing definitions involving vector orthogonality and subspace projections. It serves as a critical theoretical bridge within applied mathematics for handling overdetermined systems where exact solutions do not exist, ensuring minimal error estimation under specific convexity conditions.
The least squares method provides a mechanism to solve inconsistent linear systems $Ax = b$ by minimizing the Euclidean norm $\|Ax - b\|$ between two vectors, which corresponds to finding the orthogo…