Question 12: Suppose the temperature (T) at a point (x, y, z) in \mat…

Suppose the temperature ($T$) at a point $(x, y, z)$ in $\mathbb{R}^3$ is given by the function

$T(x, y, z) = e^{x+y+z}$

Let $T_1$, $T_2$, and $T_3$ be the tangent planes to the function $f(x, y) = \sqrt{14 - x^2 - y^2}$ at the points $(2, 3)$, $(3, 2)$, and $(3, 1)$, respectively. The set of points on each plane $T_1$, $T_2$, and $T_3$ form affine subspaces $A_1$, $A_2$, and $A_3$, respectively, of the vector space $\mathbb{R}^3$, with respect to usual addition and scalar multiplication. Let $V_1$, $V_2$, and $V_3$ denote the vector subspaces corresponding to the affine subspaces $A_1$, $A_2$, and $A_3$, respectively.

Answer the subquestions using the given information.

Let $T$ be a linear transformation between the vector subspaces $V_1$ and $V_2$. Suppose the matrix representation of $T$ with respect to the ordered bases $\beta = \{(1,0,-2),(0,1,-3)\}$ for $V_1$ and $\gamma = \{(1,0,-3),(0,1,-2)\}$ for $V_2$ is given by $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$. Then which of the following mappings can be an affine mapping $f$ between the affine subspaces $A_1$ and $A_2$, corresponding to $T$ ?