Question

A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) vertical reflection across the \(x\) axis (b) horizontal reflection across the \(y\) axis (c) diagonal reflection across the line \(y=x\)

Short answer

The matrix that performs the transformations is \( \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \).

Step-by-step solution

Matrix for Vertical Reflection Across the x-axis

A vertical reflection across the x-axis in a coordinate plane can be represented by the transformation matrix: \[ A_1 = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} \]This matrix negates the y-coordinates of any vector it is applied to.

Matrix for Horizontal Reflection Across the y-axis

A horizontal reflection across the y-axis is represented by the transformation matrix:\[ A_2 = \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \]This matrix negates the x-coordinates of any vector it is applied to.

Matrix for Diagonal Reflection Across the Line y=x

A reflection across the line \( y = x \) has the transformation matrix:\[ A_3 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \]This transformation swaps the x and y coordinates of any vector it is applied to.

Combine Transformations by Matrix Multiplication

To find the matrix \( A \) that performs all three transformations in order, we need to multiply the matrices in the order of the transformations:\[ A = A_3 \cdot A_2 \cdot A_1 \]Substituting the matrices from the previous steps, we have:\[ A = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} \]

Simplifying the Matrix Multiplication

First, compute \( A_2 \cdot A_1 \):\[ \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \]Next, compute \( A = A_3 \cdot (A_2 \cdot A_1) \):\[ \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 \cdot (-1) + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot (-1) \ 1 \cdot (-1) + 0 \cdot 0 & 1 \cdot 0 + 0 \cdot (-1) \end{bmatrix} = \begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix} \]

Resultant Transformation Matrix

The resultant matrix that performs the vertical reflection, horizontal reflection, and diagonal reflection in order is:\[ A = \begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix} \]

Key concepts

Vertical Reflection

A vertical reflection across the x-axis involves flipping any point over the x-axis. Essentially, you "mirror" the shape across the x-axis. This transformation is crucial when it comes to graphical manipulations in mathematics or computer graphics.

In terms of a matrix, the transformation is represented as follows:

• Matrix: \[ A_1 = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} \]

This matrix leaves the x-coordinates unchanged but changes the sign of the y-coordinates of any point.

Imagine plotting a point at (2, 3). After applying vertical reflection, it would be at (2, -3). The y value flips from positive to negative, reflecting the point across the x-axis.

Horizontal Reflection

Horizontal reflection happens when you flip a point across the y-axis, effectively mirroring the shape. In geometric terms, it transforms every point on the plane by changing the sign of the x-coordinate while leaving the y-coordinate intact.

The transformation can be performed using this matrix:

• Matrix: \[ A_2 = \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \]

When this transformation is applied, all x-values of the coordinates change their signs. For example, if you take a point (3, 4), after applying the horizontal reflection, it will be at (-3, 4), with the x value now negative.

Diagonal Reflection

Reflecting across the line \(y = x\) is known as diagonal reflection, which is different because it swaps the x and y values of a point. This particular reflection is useful in various applications such as solving equations or transformations in graphics.

The matrix that executes this transformation is:

• Matrix: \[ A_3 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \]

Applying this matrix swaps each point’s x and y coordinates. So a point at (5, 2) will be reflected into (2, 5) across the line \(y = x\). This swapping operation changes the orientation of objects in geometric transformations.

Matrix Multiplication

Understanding matrix multiplication is vital for performing a sequence of transformations efficiently in mathematics. When applied to transformation matrices, matrix multiplication allows us to combine multiple transformations into a single operation.

To find a matrix that reflects over the x-axis, then the y-axis, and finally across the line \(y = x\), you're essentially multiplying the transformation matrices in the specified order. The sequence of multiplication is: \[ A = A_3 \cdot A_2 \cdot A_1 \]

• Step-by-step Multiplication:
  • Multiply \(A_2 \cdot A_1\) first.
  • Then, multiply the resulting matrix by \(A_3\).

Each multiplication combines the effects of the transformations, resulting in one comprehensive matrix that applies all transformations in sequence. The final matrix derived through careful multiplication is:

• \[ \begin{bmatrix} 0 & -1 \ -1 & 0 \end{bmatrix} \]

This matrix encapsulates all specified reflections, making the process of applying them far simpler and more efficient.