Exercise 4.3.26

Answers

The unsolvable equations for $C + Dx + Ey = (0, 1, 3, 4)$ at the 4 corners are

[\begin{matrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & - 1 & 0 \\ 1 & 0 & - 1 \end{matrix}] [\begin{matrix} C \\ D \\ E \end{matrix}] = [\begin{matrix} 0 \\ 1 \\ 3 \\ 4 \end{matrix}]

Then

A^{T} A = [\begin{matrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix}]

and

A^{T} b = [\begin{matrix} 8 \\ - 2 \\ - 3 \end{matrix}]

and

[\begin{matrix} C \\ D \\ E \end{matrix}] = [\begin{matrix} 2 \\ - 1 \\ - 3 ∕ 2 \end{matrix}] .

At $x, y = 0, 0$ the best plane $2 - x - \frac{3}{2} y$ has height $C = 2 =$ average of $0, 1, 3, 4$

mathchakra

2023-09-14 08:15