Equation of a Surface relative to a basis

[shonen]

Equation of a Surface relative to the standard basis is

X1^1 + 2X2^2 + 3x^3 -4x1x2 -4x2x3 =0

Now the question ask to find the equation of above surface relative to the cordinate system with basis vectors

F1= (2/3,2/3,1/3)
F2= (1/3,-2/3,2/3)
F3= (2/3,-1/3,-2/3)

Now i found the transition matrix from the standard basis ( (1,0,0),(0,0,1),(0,0,1) ) to (F1,F2,F3) which was found to be

(2/3, 2/3, 1/3)
(1/3,-2/3, 2/3)
(2/3 -1/3,-2/3)

and got an arbitrary vector (a,b,c) in coordinate system in terms of coordinate (x1,x2,x3) relative to the standard basis

27a = 2x1 + 2x2 +x3
27b = x1 -2x2 +2x3
27bc= 2x1 -x2 -2x3

Problem is expressing the above equation of a surface in terms of a,b,and c.

 

[Stephen Tashi]

Now i found the transition matrix from the standard basis ( (1,0,0),(0,0,1),(0,0,1) ) to (F1,F2,F3) which was found to be

(2,2,1)
(1-2 3) * (1/(3)^3)
(2-1-2)

How will that matrix transform (0,0,1) to the correct representation in terms of F1,F2,F3?

 

[shonen]

How will that matrix transform (0,0,1) to the correct representation in terms of F1,F2,F3?

Let B be the matrix whose row are the standard basis, and C the matrix whose row i is the 3x1 basis vector Fi (i=1,2,3) for the coordinate system. and

p =(2/3, 2/3, 1/3)
(1/3,-2/3, 2/3)
(2/3 -1/3,-2/3)

Then

B=PT (P transpose)(C) ---(2)

Where P the transition matrix from the standard basis to C. Is it ok now ?

In the above equation I've just expressed the standard basis as a linear combination of F1,F2,F3. From definition the transition matrix from standard basis to F1,F2,F3 is one whose columns are the coordinates of the standard basis relative to F1,F2,F3.

 

[Stephen Tashi]

I think that version of $$P$$ is correct.

$$X_1 = 1 X_1 + 0 X_2 + 0 X_3 = \begin{pmatrix} 1\\0 \\0 \end{pmatrix}$$ in the standard basis. Use $$P^T$$ to transform $$X_1$$ into the $$F$$ basis. You get some vector $$\begin{pmatrix} a \\b \\c \end{pmatrix}$$ so $$X_1 = a F_1 + b F_2 + c F_3$$ and you can substitute that for $$X_1$$ in the equation for the surface. etc.