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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.

The image depicts a linear transformation T: R^2 → R^2 and asks to compute T on the vector (9, 0).

Learn how to solve a linear transformation T: R^2 → R^2 applied to the vector (9, 0) using matrix multiplication and scalar operations. This problem covers key concepts in linear algebra, including matrix transformation, vector addition, and scalar multiplication. Suitable for students studying linear algebra at the undergraduate level.

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