QuestionConsider the weighted Euclidean inner product defined by $<u, v>=7 u_{1} v_{1}+6 u_{2} v_{2}+4 u_{3} v_{3}$. The generating matrix for this inner product is given by $A=$ $$\left[\begin{array}{ccc} \sqrt{7} & 0 & 0 \\ 0 & \sqrt{6} & 0 \\ 0 & 0 & 2 \end{array}\right]$$ True False

Studdy Solution
Verify if the given matrix $A$ satisfies the condition for the generating matrix. We need to check if:
$$<u, v> = (Au) \cdot (Av)$$
Given $A = \left[\begin{array}{ccc} \sqrt{7} & 0 & 0 \\ 0 & \sqrt{6} & 0 \\ 0 & 0 & 2 \end{array}\right]$, compute $Au$ and $Av$:
$$Au = \left[\begin{array}{ccc} \sqrt{7} & 0 & 0 \\ 0 & \sqrt{6} & 0 \\ 0 & 0 & 2 \end{array}\right] \left[\begin{array}{c} u_1 \\ u_2 \\ u_3 \end{array}\right] = \left[\begin{array}{c} \sqrt{7} u_1 \\ \sqrt{6} u_2 \\ 2 u_3 \end{array}\right]$$
$$Av = \left[\begin{array}{ccc} \sqrt{7} & 0 & 0 \\ 0 & \sqrt{6} & 0 \\ 0 & 0 & 2 \end{array}\right] \left[\begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array}\right] = \left[\begin{array}{c} \sqrt{7} v_1 \\ \sqrt{6} v_2 \\ 2 v_3 \end{array}\right]$$
Calculate the standard Euclidean inner product $(Au) \cdot (Av)$:
$$(\sqrt{7} u_1)(\sqrt{7} v_1) + (\sqrt{6} u_2)(\sqrt{6} v_2) + (2 u_3)(2 v_3)$$
$$= 7 u_1 v_1 + 6 u_2 v_2 + 4 u_3 v_3$$
This matches the given weighted inner product $<u, v> = 7 u_1 v_1 + 6 u_2 v_2 + 4 u_3 v_3$.
The statement is:
$$\text{True}$$