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Figure 14) The affine hull of three noncollinear points in any dimension is that unique plane containing the points, and so on. 13 The popular term affine subspace is an oxymoron. 3. 1) in R3 whose boundary is drawn truncated; constructed using A ∈ R3×3 and C = 0 in (236). 1), entire boundary can be constructed from an aggregate of rays emanating exclusively from the origin. 3); they are conically, affinely, and linearly independent for this cone. Because this set is polyhedral, exposed directions are in one-to-one correspondence with extreme directions; there are only three.

3. HULLS A affine hull (drawn truncated) C convex hull K conic hull (truncated) range or span is a plane (truncated) R Figure 14: Given two points in Euclidean vector space of any dimension, their various hulls are illustrated. Each hull is a subset of range; generally, A , C, K ⊆ R . ) 72 CHAPTER 2. 4 (81) Vertex-description The conditions in (66), (71), and (79) respectively define an affine, convex, and conic combination of elements from the set or list. Whenever a Euclidean body can be described as some hull or span of a set of points, then that representation is loosely called a vertex-description.

2 57 Vectorized-matrix inner product Euclidean space Rn comes equipped with a linear vector inner-product ∆ y , z = y Tz (25) We prefer those angle brackets to connote a geometric rather than algebraic perspective. Two vectors are orthogonal (perpendicular ) to one another if and only if their inner product vanishes; y⊥z ⇔ y, z = 0 (26) A vector inner-product defines a norm y ∆ 2 = yT y , y 2 =0 ⇔ y=0 (27) When orthogonal vectors each have unit norm, then they are orthonormal. 10] ∆ y , ATz = Ay , z (28) The vector inner-product for matrices is calculated just as it is for vectors; by first transforming a matrix in Rp×k to a vector in Rpk by concatenating its columns in the natural order.

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A brill - noether theory for k-gonal nodal curves by Ballico E.

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