Linear combination of vector
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This document defines and provides examples of linear combinations of vectors. It discusses two types of linear combinations: affine and convex. An affine combination requires that the coefficients of the vectors add up to 1. A convex combination is a restriction of an affine combination where the coefficients must also be non-negative and between 0 and 1. Examples are provided to illustrate affine and convex combinations.
Linear combination of vector
- 1. Computer Graphics (Assignment) By: Farwa Abdul Hannan (12 – CS – 13) Linear Combination of Vector: Definition: A linear combination of the m vectors V1, V2, … , Vm is a vector of the form W = A1 V1+ A2 V2 +... + Am Vm (where A1, A2, … , Am are scalars). Example: The linear combination of two scalars A1 and A2 with two vectors V1 and V2 respectively forms a vector V that is 2(3, 4,-1) + 6(-1, 0, 2) forms the vector (0, 8, 10). Types: There are two types of linear combination of vectors Affine Convex Affine combination of vectors: A linear combination is an affine combination if the coefficients A1, A2, . . . , Am add up to 1. Thus the linear combination in A1 + A2 + ... + Am is affine if: A1 + A2 + ... + Am = 1 Example: 3 a + 2 b - 4 c is an affine combination of a, b, and c, but 3 a + b - 4 c is not. Convex combination of vectors: A convex combination is a restriction to affine combination of vectors. A combination of vectors is convex if the sum of all the coefficients is 1, and each coefficient must also be non- negative and mathematically it is A1 + A2 + ... + Am = 1 Where Ai 0, for i = 1,…, m.. As a consequence all Ai must lie between 0 and 1. Example: 0.3a + 0.7b is a convex combination of a and b, but 1.8a - 0.8b is not. The later one is not a convex combination of vectors because the coefficient of first term is 1.8 which doesn’t match the condition of convex combination which states that all the coefficients must lie between 0 and 1.