Orthogonal projection onto a subspace $W$ represents the vector in $W$ that is closest to a given target vector $\mathbf{y}$, thereby providing the best linear approximation within that domain. The theory defines an orthogonal decomposition where any vector $\mathbf{y} \in \mathbb{R}^n$ can be expressed as the sum of its projection onto a subspace generated by $p$ basis vectors and a residual component strictly belonging to the orthogonal complement $W^\perp$. This mechanism is fundamental in linear algebra for solving inconsistent systems, minimizing least-squares errors, and characterizing vector spaces through their subspaces and complements.
Orthogonal projection onto a subspace $W$ represents the vector in $W$ that is closest to a given target vector $\mathbf{y}$, thereby providing the best linear approximation within that domain. The t…