Given the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $T(X) = AX$, where:

• $X = \begin{pmatrix} x \\ y \end{pmatrix}$
• The matrix $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

we need to determine what $T(x, y)$ is in terms of $x$ and $y$.

Step 1: Apply the transformation

The transformation is given by:

$T \left( \begin{pmatrix} x \\ y \end{pmatrix} \right) = A \begin{pmatrix} x \\ y \end{pmatrix}$

So we multiply $A$ by $X$:

$A \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$

Step 2: Perform the matrix multiplication

Carrying out the multiplication:

$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \cdot x + 1 \cdot y \\ 1 \cdot x + 0 \cdot y \end{pmatrix} = \begin{pmatrix} y \\ x \end{pmatrix}$

Conclusion

The result is:

$T(x, y) = (y, x)$

Correct Answer

• Option 3: $T(x, y) = (y, x)$

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. What does the matrix $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ represent geometrically in $\mathbb{R}^2$?
2. How would $T$ change if the matrix $A$ had different values?
3. What are the eigenvalues and eigenvectors of the matrix $A$?
4. How do you determine if a transformation is a reflection, rotation, or scaling in $\mathbb{R}^2$?
5. What is the determinant of $A$, and what does it tell us about the transformation $T$?

Tip: When applying a matrix transformation, perform matrix multiplication carefully and check each term, as this defines the resulting transformation accurately.