Linear Combination of Vector: introduction, examples and types (Computer Graphics)

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.