Find the coordinate vector of x with respect to the basis B

Question

Find the coordinate vector of x=[-1,2,4] with respect to the basis B={[1,4,6],[0,1,-4],[0,0,1]}

 

|x|B=

Answer 1

We want [a,b,c] with a, b, and c satisfying

 

[-1,2,4] = a[1,4,6] + b[0,1,-4] + c[0,0,1]

 

 

Equating components:

 

-1 = a

 

2 = 4a + b = -4 + b   →   b = 6

 

4 = 6a - 4b + c = -6 - 24 + c   →   c = 34

 

[-1,6,34] is the coordinate vector with respect to basis B.