Discover Linear Algebra

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