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\)
Section 38.2 Terminology and notation
Where they appear in the following definitions, \(\uvec{v}\) represents a vector in an inner product space \(V\) and \(U\) represents a subspace of \(V\text{.}\)
orthogonal projection of \(\uvec{v}\) onto \(U\)
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 \(\uvec{v}\) orthogonal \(U\)
the vector
\(\proj_{\orthogcmp{U}} \uvec{v} = \uvec{v} - \proj_U \uvec{v}\)
best approximation to \(\uvec{v}\) within \(U\)
the orthogonal projection
\(\proj_U \uvec{v}\)
distance between \(\uvec{v}\) and \(U\)
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 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 possibly nonsquare matrix \(A\) )
the matrix
\(\inv{(\utrans{A} A)} \utrans{A}\text{,}\) in the case that
\(\utrans{A} A\) is invertible
