Question 20: Let T denote the linear transformation from V_1 to V_2, …

• Consider the function $f : \mathbb{R}^2 \to \mathbb{R}$ defined as

  $f(x, y) = x^2y + xy^2$

• The tangent plane to the function $f$ at the point (1,1) is given by $T_1$. Let $A_1$ denote the affine subspace of $\mathbb{R}^3$ formed by the set of points on $T_1$. Let $V_1$ denote the corresponding vector subspace associated with $A_1$.

• Let $A$ denote the Hessian matrix of the function $f$ at the point $(1, 1)$.

• Consider the affine subspace of $\mathbb{R}^3$ given by

  $A_2 = \{(x, y, z) \mid x - y + z = 1, \text{ where } x, y, z \in \mathbb{R}\}$

  Let $V_2$ denote the corresponding vector subspace associated with $A_2$.

Using the above information answer the given subquestions.

Let $T$ denote the linear transformation from $V_1$ to $V_2$, whose matrix representation with respect to the ordered bases $\beta$ and $\gamma$ (mentioned in the previous questions 19 & 20 ), respectively for $V_1$ and $V_2$ is the identity matrix, i.e.,$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Which of the following functions denotes an affine mapping from $A_1$ to $A_2$ corresponding to $T$?