LINEAR ALGEBRA AFFINE COMBINATIONS Let v1 3 5 v2 3 2 v3 2

****LINEAR ALGEBRA, AFFINE COMBINATIONS****

Let v1 = (3, 5), v2 = (3, 2), v3 = (2, 3), and v4 = (6, 8). Then a convex combination of (1, 5) is given by

(1, 5) = (5/12)v₁ + (1/12)v₂ + (3/12)v₃ + (3/12)v₄.

An anely dependent relation of v1, v2, v3 and v4 is given by

3v₁ + 11v₂ 15v₃ + v₄ = 0.
Express (1, 5) as a convex combination of 3 of the 4 vectors. Find two distinct representations. Reduce your coecients to just a fraction (not a sum/dierence of fractions).

Solution

Given convex cobination of (-1, 5) is (-1,5) = (5/12)v1+(1/12)v2+(3/12)v3+(3/12)v4 -----------------------(1)

also given relation of v1, v2, v3 and v4 as 3v1 + 11v2 15v3 + v4 = 0-----------------------------------------(2)

from (2), we can write 15v3.=3v1 + 11v2 + v4. This implies v3.=(3/15) v1 + (11/15) v2 +(1/15) v4------------------(3)

Substitute v3 value from (3) into (1) we get

(-1,5) = (5/12)v1+(1/12)v2+(3/12) [(3/15) v1 + (11/15) v2 +(1/15) v4] +(3/12)v4

=[(5/12)+(1/20)]v1+[(1/12)+(11/60)]v2+[(3/12)+(1/60)]v4

(-1, 5)=(7/15)v1+(4/15)v2+(4/15)v4