The transformation $T$ is defined by:

$T(x_1, x_2) = \left( \frac{x_1 + x_2}{2}, \frac{x_1 + x_2}{2} \right)$

We want to express this linear transformation in matrix form. To do this, we apply $T$ to the standard basis vectors of $\mathbb{R}^2$:

1. Apply $T$ to $(1, 0)$: $T(1, 0) = \left( \frac{1 + 0}{2}, \frac{1 + 0}{2} \right) = \left( \frac{1}{2}, \frac{1}{2} \right)$
2. Apply $T$ to $(0, 1)$: $T(0, 1) = \left( \frac{0 + 1}{2}, \frac{0 + 1}{2} \right) = \left( \frac{1}{2}, \frac{1}{2} \right)$

These results form the columns of the transformation matrix. So, the matrix $A$ representing $T$ is:

$A = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}$

Thus, the correct option is:

Oa. $\begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}$

Would you like further details on the transformation or how matrices represent linear transformations?

Here are 5 related questions to expand on this:

1. How do we find the matrix of a linear transformation in general?
2. What are the properties of the matrix found for this transformation?
3. How do we verify if a given transformation is linear?
4. How does this transformation affect vectors geometrically in $\mathbb{R}^2$?
5. What would happen if the transformation was not linear—how would that affect the matrix representation?

Tip: For any linear transformation, applying it to the standard basis vectors helps derive the corresponding matrix.