Find the coordinate vector

Question

of x= [-2 -2 -3] with respect to the basis B=([1 6 2], [0 1 -7], [0 0 1]) or R3

[X]_B=

Answer 1

Let us call [1 6 2], [0 1 -7], [0 0 1] by b₁, b₂, b₃, respectively. To find the coordinate vector [a b c] of x in Basis B, we need to find a,b,c, satifying

x = ab₁+bb₂+cb₃.

So we have a system of equations

-2 = a + 0 + 0

-2 = 6a + b + 0

-3 = 2a - 7b + c.

We have a=-2, b=10, c = 71.