Question

Let (x1;x2;x3) and (y1;y2;y3) represent the coordinates with respect to the bases
B1 = f(1;0;0); (0;1;0); (0;0;1)g, B2 = f(1;0;0); (0;1;2); (0;2;1)g. If
Q(X) = x2
1+2x1x2+2x2x3+x2
2+x2
3, find the representation of Q in terms of
(y1;y2;y3).

Answer 1

Answer on Question #79908 – Math – Linear Algebra

Question

Let $(x_1; x_2; x_3)$ and $(y_1; y_2; y_3)$ represent the coordinates with respect to the bases $B_1 = \{(1; 0; 0), (0; 1; 0), (0; 0; 1)\}$, $B_2 = \{(1; 0; 0), (0; 1; 2), (0; 2; 1)\}$.

If

$$Q(X) = x_1^2 + 2x_1x_2 + 2x_2x_3 + x_2^2 + x_3^2,$$

find the representation of $Q$ in terms of $(y_1; y_2; y_3)$.

Solution

$$A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{bmatrix}$$ $$P = P_{B_1 \to B_2} = \begin{bmatrix} 1 & 0 & 0 & | & 1 & 0 & 0 \\ 0 & 1 & 2 & | & 0 & 1 & 0 \\ 0 & 2 & 1 & | & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & | & 1 & 0 & 0 \\ 0 & 1 & 2 & | & 0 & 1 & 0 \\ 0 & 0 & -3 & | & 0 & -2 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & | & 1 & 0 & 0 \\ 0 & 1 & 0 & | & 0 & -1/3 & 2/3 \\ 0 & 0 & 1 & | & 0 & 2/3 & -1/3 \end{bmatrix}$$ $$A_2 = P^T A P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1/3 & 2/3 \\ 0 & 2/3 & -1/3 \end{bmatrix} \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1/3 & 2/3 \\ 0 & 2/3 & -1/3 \end{bmatrix} = \begin{bmatrix} 1 & -1/3 & 2/3 \\ -1/3 & 1/9 & 1/9 \\ 2/3 & 1/9 & 1/9 \end{bmatrix}$$

Answer:

$$Q(Y) = y_1^2 + 1/9 y_2^2 + 1/9 y_3^2 - 2/3 y_1 y_2 + 4/3 y_1 y_3 + 2/9 y_2 y_3.$$

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