Find the coordinate vector of 𝑥⃗ =

Question

Find the coordinate vector of 𝑥⃗ = < 1,2,-3 > with respect to the basis

𝐵 = {<1,4,4> , <0,1,-8> , <0,0,1>} for R³

[𝑥⃗ ]_B = < _ , _ , _ >

Answer 1

Denote the coordinate vector by <a, b, c >

Then, a<1, 4, 4> + b<0, 1, -8> + c<0, 0, 1> = <1, 2, -3>

Equating components, we have:

a + 0 + 0 = 1

4a + b + 0 = 2

4a - 8b + c = -3

So, a = 1, b = -2, and c = -23

The desired coordinate vector is < 1, -2, -23 >