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**Additional resources for Algebraic Geometry Santa Cruz 1995, Part 2**

**Sample text**

65) A half space is defined in a similar manner. 66) where f : X → R is affine and c ∈ R. If x1 , . . , xn are coordinates for points in X with respect to some coordinate system, then a half space is given by an equation of the form a1 x1 + · · · + an xn ≥ c. 30 A subset C ⊆ X of an affine space is called convex if for each pair of points in C the line segment between the points are in C, see Fig. 8. 31 Let A ⊆ X be an arbitrary subset of an affine space X. The convex hull of A is the smallest convex set containing A and is denoted CH (A).

E −1 }⊥ ∧ e = 1} and that L(e ) e realizes the maximum. The basis for V is simply given as f = L(e ) when L(e ) = 0. If this gives f1 , . . , fk then the rest of the basis vectors are chosen as an orthonormal basis for span{f1 , . . , fk }⊥ . In terms of matrices it has the following formulation. 12 Let A be an m × n matrix and let k = min{m, n}. Then A can be decomposed as A = UΣVT , where U is an orthogonal m × m matrix, V is an orthogonal n × n matrix, and Σ is a diagonal matrix with non zero elements σ1 ≥ σ1 ≥ · · · ≥ σk ≥ 0 in the diagonal.

70 are metric spaces. 73. 22 Let A be a real n × m matrix of rank m ≤ n, let x ∈ Rm , and let b ∈ Rn . What is the solution to min f (x)| x x =1 , where f (x) = xT AT Ax? , AT A = 5 −1 −1 5 . 23 What are the solutions to max f (x), x where f (x) = bT x ? 24 What geometric object do the points x ∈ R3 , fulfilling the equation nT x = α, describe? Here n ∈ R3 and α ∈ R. Please explain. References 1. : Linear Algebra and Its Applications, 4th edn. Brooks Cole (2006) 2. : Fundamental Concepts in Modern Analysis.