Let a1 4 4 5 a2 4 1 1 and b 0 25 5 20 Is b a linear combi

Let a_1 = [4 -4 -5], a_2 = [-4 -1 1], and b = [0 -25 -5 -20] Is b a linear combination of a_1 and a_2 ? b is not a linear combination. Yes b is a linear combination. We cannot tell if b is a linear combination. Either fill in the coefficients of the vector equation, or enter \"NONE\" if no solution is possible.

Solution

If b isa linear combination of a1 and a2 , we can write:

xa1 +ya2 = b

if x = 5 and y=5 we get:

5a1 +5a2 =b

5(4 -4 -5 ) +5(-4 -1 1) = ( 0 -25 -20)

(0 -25 20) = ( 0 -25 -20)

So, b is a linear combination of a1 and a2

b = 5a1 +5a2