Section 38.2 Terminology and notation
- orthogonal projection (of a vector \(\uvec{v}\) onto a subspace \(U\))
-
the vector \(\uvec{u}\) in \(U\) in the unique decomposition
\begin{equation*} \uvec{v} = \uvec{u} + \uvec{u}' \end{equation*}of \(\uvec{v}\) into the sum of a vector in \(U\) and a vector in \(\orthogcmp{U}\text{;}\) sometimes called the vector component of \(\uvec{v}\) parallel to \(U\)
- vector component of a vector \(\uvec{v}\) orthogonal to a subspace \(U\)
the vector \(\proj_{\orthogcmp{U}} \uvec{v} = \uvec{v} - \proj_U \uvec{v}\)
- best approximation (to a vector \(\uvec{v}\) from within a subspace \(U\))
the orthogonal projection \(\proj_U \uvec{v}\)
- distance between a vector \(\uvec{v}\) and a subspace \(U\)
the smallest possible value of \(\dist (\uvec{v}, \uvec{u})\) amongst all vectors \(\uvec{u}\) in \(U\text{,}\) denoted \(\dist (\uvec{v}, U)\)
- normal system (associated to an inconsistent system \(A \uvec{x} = \uvec{b}\))
the transformed system \(\utrans{A} A \uvec{x} = \utrans{A} \uvec{b}\)
- least-squares solution (for an inconsistent system)
a solution to the normal system associated to the inconsistent system
- pseudo-inverse matrix (of a possibly non-square matrix \(A\))
the matrix \(\inv{(\utrans{A} A)} \utrans{A}\text{,}\) in the case that \(\utrans{A} A\) is invertible