Let 𝑇:𝑅2→𝑅2

 be a linear transformation defined by T(X)=AX where X=

(𝑥 ,𝑦)

and the matrix is 𝐴=([0 1], [1 0])

. Then T is

Group of answer choices


T(x,y)=(y,-x)

T(x,y)=(-y,x)

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

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

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.

Let T: R^2 → R^2 be a linear transformation defined by T(X) = AX where X = (x, y) and the matrix is A = ([0 1], [1 0]). Then T is

Group of answer choices

T(x,y)=(y,-x)

T(x,y)=(-y,x)

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

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

Explore how to determine the result of a linear transformation T: R^2 → R^2 with matrix A = [[0, 1], [1, 0]] on vector X = (x, y). Learn key concepts in linear algebra, including matrix multiplication and properties of linear transformations. Suitable for undergraduate linear algebra students.

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