Find the coordinate vector of x=[-4,1,-2] with respect to the basis B=([1,7,7], [0,1,-6], [001]) for R^3

Question

Find the coordinate vector of x=[-4,1,-2] with respect to the basis B=([1,7,7], [0,1,-6], [001]) for R^3

X_B=[?,?,?]

Answer 1

Find a, b, and c so that a(1,7,7) + b(0,1,-6) + c(0,0,1) = (-4,1,-2)

a + 0 + 0 = -4

7a + b + 0 = 1

7a - 6b + c = -2

From the first equation, a = -4

So, using the second equation, -28 + b = 1. Therefore, b = 29.

Substitute into the third equation to get -28 - 174 + c = -2. So, c = 200

X_B = (-4, 29, 200)