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.