The image depicts a mathematical problem involving a linear transformation $T : \mathbb{R}^2 \to \mathbb{R}^2$. Here's a breakdown of what I can infer:

1. The linear transformation is defined by applying the matrix transformation to a 2D vector $\begin{pmatrix} x \\ y \end{pmatrix}$ and involves matrix addition and scalar multiplication.
2. The expression seems to involve the computation of $T$ on the vector $\begin{pmatrix} 9 \\ 0 \end{pmatrix}$.

I will now reconstruct the linear transformation rule and help you solve the specific case for $\begin{pmatrix} 9 \\ 0 \end{pmatrix}$.

Let me break it down and solve it.The linear transformation $T$ applied to the vector $\begin{pmatrix} 9 \\ 0 \end{pmatrix}$ results in:

$T\left( \begin{pmatrix} 9 \\ 0 \end{pmatrix} \right) = \begin{pmatrix} 27 \\ 9 \end{pmatrix}$

Would you like further details or have any questions?

Here are some related questions to explore:

1. What are the general steps to define a linear transformation in matrix form?
2. How do scalar multiplication and matrix addition affect linear transformations?
3. What are the conditions for a function to be considered a linear transformation?
4. How can we express a linear transformation in terms of a standard matrix?
5. How would $T$ change if we input a different vector, say $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$?

Tip: When working with linear transformations, try expressing them as matrix multiplications to make calculations easier.