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!