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To find the matrix representation of the linear operator \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( T(x, y) = (x + 2y, 3x + y) \), we need to express this transformation in terms of the standard basis vectors \( \mathbf{e}_1 = (1, 0) \) and \( \mathbf{e}_2 = (0, 1) \) of \( \mathbb{R}^2 \).

### Step-by-Step:

1. **Action of \( T \) on the first standard basis vector**:
   - \( T(1, 0) \): 
     \[
     T(1, 0) = (1 + 2(0), 3(1) + 0) = (1, 3)
     \]
   So, the image of \( \mathbf{e}_1 \) under \( T \) is \( (1, 3) \).

2. **Action of \( T \) on the second standard basis vector**:
   - \( T(0, 1) \):
     \[
     T(0, 1) = (0 + 2(1), 3(0) + 1) = (2, 1)
     \]
   So, the image of \( \mathbf{e}_2 \) under \( T \) is \( (2, 1) \).

3. **Form the matrix**:
   The matrix representation of \( T \) is given by placing the images of the standard basis vectors as columns of the matrix. Thus, the matrix is:
   \[
   [T] = \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix}
   \]

### Conclusion:
The correct matrix is \( \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix} \), which corresponds to the first option in the provided choices.

Would you like more details or have any questions? 

Here are some related questions for further exploration:

1. How would you find the matrix representation of a linear transformation in \( \mathbb{R}^3 \)?
2. How can you verify that a given transformation is linear?
3. What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix} \)?
4. How does the choice of basis affect the matrix representation of a linear transformation?
5. Can you express the inverse of this linear transformation?

**Tip**: To check if your matrix representation is correct, you can apply the matrix to the standard basis vectors and see if the results match the transformation's definition!

The matrix representation of the linear operator T: R^2 -> R^2 defined by T(x,y) = (x + 2y, 3x + y) for all (x,y) ∈ R^2, with respect to the standard basis is ____.

Find the matrix representation of the linear operator T: R^2 -> R^2 defined by T(x, y) = (x + 2y, 3x + y). This problem covers key concepts in linear algebra, including matrix representation and linear transformations. Suitable for students in Grades 10-12.

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