Find the coordinate matrix of x relative to the given basis

Find the coordinate matrix of x relative to the given basis B\', where B\' = {(1,2,3),(1,2,0),(0,-6,2)}. X= (3,-3,0)

Solution

Let coordinate vector entries of x are x₁, x₂, x₃

such that they meet the following criterion:

x₁ (1,2,3) +  x₂ (1,2,0) +  x₃ (0,-6,2) = (3,-3,0)

Solving this, we get

x₁ + x₂ + 0 = 3 ... (1)

2x₁ + 2x₂ - 6x₃ = -3 ... (2)

3x₁ + 0 + 2x₃ = 0 ... (3)

Multiplying equation (1) by 2 , we get

2x₁ + 2x₂ = 6 ... (4)

Subtracting equation (2) from equation (4), we get

6x₃ = 6 + 3 = 9

So, x₃ = 3/2

Putting x₃ = 3/2 in equation (3), we get

3x₁ + 3 = 0

So, x₁ = -1

Putting x₁ = -1 in equation (1), we get

-1 + x₂ = 3

So, x₂ = 4

(x₁, x₂, x₃) = (-1, 4 , 3/2)

Thus, the coordinate matrix of x is (-1, 4 ,3/2).