Understanding the Composite Linear Transformation

The question asks for the standard matrix of a linear transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\) which is formed by performing two successive reflections:

1. First, reflecting points through the vertical axis (the \(x_2\)-axis, where \(x_1 = 0\)). Let's call this transformation \(T_1\).
2. Second, reflecting the result through the line \(x_2 = x_1\). Let's call this transformation \(T_2\).

The total transformation \(T\) is the composition of \(T_1\) and \(T_2\), meaning \(T(\mathbf{v}) = T_2(T_1(\mathbf{v}))\) for any vector \(\mathbf{v}\) in \(\mathbb{R}^2\). The standard matrix of a composite transformation is found by multiplying the standard matrices of the individual transformations in the reverse order of application. If \(M_1\) is the standard matrix for \(T_1\) and \(M_2\) is the standard matrix for \(T_2\), then the standard matrix for \(T = T_2 \circ T_1\) is \(M = M_2 M_1\).

Step 1: Finding the Standard Matrix for Reflection Through the Vertical Axis (\(T_1\))

A reflection through the vertical axis (\(x_2\)-axis) transforms a point \(\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\) to \(\begin{bmatrix} -x_1 \\ x_2 \end{bmatrix}\). To find the standard matrix \(M_1\), we apply this transformation to the standard basis vectors \(\mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\) and \(\mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\).

• \(T_1(\mathbf{e}_1) = T_1(\begin{bmatrix} 1 \\ 0 \end{bmatrix}) = \begin{bmatrix} -1 \\ 0 \end{bmatrix}\)
• \(T_1(\mathbf{e}_2) = T_1(\begin{bmatrix} 0 \\ 1 \end{bmatrix}) = \begin{bmatrix} -0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)

The standard matrix \(M_1\) is formed by using the results as columns:

So, the standard matrix for \(T_1\) is \(M_1 = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\).

Step 2: Finding the Standard Matrix for Reflection Through the Line \(x_2 = x_1\) (\(T_2\))

A reflection through the line \(x_2 = x_1\) swaps the coordinates of a point. That is, \(\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\) transforms to \(\begin{bmatrix} x_2 \\ x_1 \end{bmatrix}\). To find the standard matrix \(M_2\), we apply this transformation to the standard basis vectors.

• \(T_2(\mathbf{e}_1) = T_2(\begin{bmatrix} 1 \\ 0 \end{bmatrix}) = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\)
• \(T_2(\mathbf{e}_2) = T_2(\begin{bmatrix} 0 \\ 1 \end{bmatrix}) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\)

The standard matrix \(M_2\) is formed by using the results as columns:

So, the standard matrix for \(T_2\) is \(M_2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\).

Step 3: Finding the Standard Matrix of the Composite Transformation \(T\)

The transformation \(T\) first applies \(T_1\) and then \(T_2\). The standard matrix \(M\) of the composite transformation \(T = T_2 \circ T_1\) is the product \(M_2 M_1\).

We need to compute the matrix product:

\(M = M_2 M_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)

Performing the matrix multiplication:

\(M = \begin{bmatrix} (0)(-1) + (1)(0) & (0)(0) + (1)(1) \\ (1)(-1) + (0)(0) & (1)(0) + (0)(1) \end{bmatrix}\)

\(M = \begin{bmatrix} 0 + 0 & 0 + 1 \\ -1 + 0 & 0 + 0 \end{bmatrix}\)

\(M = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)

Thus, the standard matrix of the transformation \(T\) is \(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\).

Conclusion

The standard matrix for the transformation \(T\) which first reflects points through the vertical axis and then reflects points through the line \(x_2 = x_1\) is \(\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\).

Revision Table: Standard Matrices

Additional Information: Composite Transformations

When a linear transformation \(T\) is a sequence of linear transformations \(T_1, T_2, \dots, T_k\), applied in that order (first \(T_1\), then \(T_2\), ..., then \(T_k\)), the standard matrix of the composite transformation \(T = T_k \circ \dots \circ T_2 \circ T_1\) is the product of the standard matrices of the individual transformations in the reverse order of application:

\(M = M_k M_{k-1} \dots M_2 M_1\)

Here, \(M_i\) is the standard matrix of the transformation \(T_i\). The order is crucial because matrix multiplication is not commutative in general. In this problem, \(T = T_2 \circ T_1\), so the matrix is \(M_2 M_1\).

Each linear transformation in \(\mathbb{R}^2\) has a unique \(2 \times 2\) standard matrix. This matrix completely describes the transformation and allows us to calculate the transformed vector by simple matrix-vector multiplication: \(T(\mathbf{v}) = M\mathbf{v}\).