Exercise 21.10 [10x10-exercise]

Compute the true utility function and the best linear approximation in $x$ and $y$ (as in Equation (4x3-linear-approx-equation)) for the following environments:

1. A ${10}\times {10}$ world with a single $+1$ terminal state at (10,10).

2. As in (a), but add a $-1$ terminal state at (10,1).

3. As in (b), but add obstacles in 10 randomly selected squares.

4. As in (b), but place a wall stretching from (5,2) to (5,9).

5. As in (a), but with the terminal state at (5,5).

The actions are deterministic moves in the four directions. In each case, compare the results using three-dimensional plots. For each environment, propose additional features (besides $x$ and $y$) that would improve the approximation and show the results.