v2010.10.26 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 55BR 3PTR 3PT(B)xPTxFigure 19: Linear noninjective mapping PTx=A † Ax : R 3 → R 3 ofthree-dimensional Euclidean body B has affine dimension 2 under projectionon rowspace of fat full-rank matrix A∈ R 2×3 . Set of coefficients of orthogonalprojection T B = {Ax |x∈ B} is isomorphic with projection P(T B) [sic].Any linear injective transformation on Euclidean space is uniquelyinvertible on its range. In fact, any linear injective transformation has arange whose dimension equals that of its domain. In other words, for anyinvertible linear transformation T [ibidem]dim dom(T) = dim R(T) (49)e.g., T represented by skinny-or-square full-rank matrices. (Figure 18) Animportant consequence of this fact is:Affine dimension, of any n-dimensional Euclidean body in domain ofoperator T , is invariant to linear injective transformation.2.2.1.2 Noninjective linear operatorsMappings in Euclidean space created by noninjective linear operators can becharacterized in terms of an orthogonal projector (E). Consider noninjectivelinear operator Tx =Ax : R n → R m represented by fat matrix A∈ R m×n(m< n). What can be said about the nature of this m-dimensional mapping?