Question

Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases B1={(1,0,0),(0,1,0),(0,0,1)}, B2={(1,0,0),(0,1,2),(0,2,1)}. If Q(x)=x1^2-2x1x2+4x2x3+x2^2+x3^2 , find the representation of Q in terms of (y1,y2,y3).

Answer 1

The following equalities are true:

$$(1, 0, 0)=1*(1, 0, 0)+0*(0, 1, 0)+0*(0, 0, 1)$$ $$(0, 1, 2)=0*(1, 0, 0)+1*(0, 1, 0)+2*(0, 0, 1)$$ $$(0, 2, 1)=0*(1, 0, 0)+2*(0, 1, 0)+1*(0, 0, 1)$$ $$C=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}$$ $$C^T=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}$$

$$Q(X)={x_1}^2+2x_1x_2+4x_2x_3+{x_2}^2+{x_3}^2$$ $$A=\begin{pmatrix} 1 & 1 & 0 \\ 1 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}$$

Then the matrix of the form Q in the base B₂ is equal to

$$B=C^TAC=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}\begin{pmatrix} 1 & 1 & 0 \\ 1 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}$$

$$B=\begin{pmatrix} 1 & 1 & 2 \\ 1 & 5 & 4 \\ 0 & 4 & 5 \end{pmatrix}\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}=\begin{pmatrix} 1 & 1 & 2 \\ 1 & 13 & 14 \\ 2 & 14 & 13 \end{pmatrix}$$

So the form Q in the base B2 will

$$Q(Y)={y_1}^2+13{y_2}^2+13{y_3}^2+2y_1y_2+4y_1y_3+28y_2y_3$$