Question

A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) horizontal stretch by a factor of 3 (b) reflection across the line \(y=x\)

Short answer

The matrix is \(A = \begin{bmatrix} 0 & 1 \\ 3 & 0 \end{bmatrix}\).

Step-by-step solution

Understand the Transformations

We need to determine the sequence of transformations to be applied. First, we'll apply a horizontal stretch by a factor of 3, followed by a reflection across the line \(y=x\).

Matrix for Horizontal Stretch

The matrix for a horizontal stretch by a factor of 3 is given by:\[A_1 = \begin{bmatrix} 3 & 0 \ 0 & 1 \end{bmatrix}.\]This matrix increases the x-coordinate by a factor of 3 while keeping the y-coordinate unchanged.

Matrix for Reflection Across the Line y=x

The matrix for reflection across the line \(y=x\) is:\[A_2 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}.\]This matrix swaps the x and y coordinates.

Calculate the Resulting Transformation Matrix

To find the resulting transformation matrix \(A\), multiply the matrices from the previous steps in the order of transformations (first horizontal stretch, then reflection):\[A = A_2 \times A_1 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 3 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \ 3 & 0 \end{bmatrix}.\]Here, each element of the resulting matrix is calculated by taking the dot product of rows from \(A_2\) and columns from \(A_1\).

Verify the Final Matrix

Verify if the resulting matrix performs the given transformations. The calculated matrix \(A = \begin{bmatrix} 0 & 1 \ 3 & 0 \end{bmatrix}\) swaps and stretches the coordinates as required, confirming that the transformations are correctly represented.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear transformations, essential for combining multiple transformations into a single operation. It allows us to apply several transformations in sequence by multiplying their corresponding matrices. This process involves taking the dot product of rows from the first matrix and columns from the second matrix.

In the exercise, you have two matrices:

• Matrix for horizontal stretch: \[\begin{bmatrix} 3 & 0 \0 & 1 \end{bmatrix}\]
• Matrix for reflection across the line \(y=x\): \[\begin{bmatrix}0 & 1 \1 & 0\end{bmatrix}\]

To find the combined effect of these transformations, the matrices are multiplied in the order they are applied. The resulting matrix \[\begin{bmatrix}0 & 1 \3 & 0\end{bmatrix}\]is obtained by matrix multiplication, confirming both transformations (reflection and stretch) are represented. Remember, the order of multiplication matters, as it determines how transformations are applied.

Geometric Transformations

Geometric transformations involve changing the size, position, or orientation of a figure in space. In mathematics, these transformations can be represented by matrices, allowing easy manipulation of shapes in coordinate systems.

Consider a few common geometric transformations:

• Translation: Moving a shape without changing its size or direction. Matrices can add the necessary shift to the coordinates.
• Scaling: Changing the size of an object, such as increasing or decreasing length by multiplying coordinates according to scale factors.
• Rotation: Turning a shape around a point. The respective matrix will rotate the coordinates based on a specified angle.
• Reflection: Flipping an object over a line, indicated by specific matrices which swap coordinates.

In the given exercise, you're dealing with a horizontal stretch and a reflection. Using matrices allows these transformations to be represented accurately, combining them efficiently through multiplication to achieve the desired effect on coordinates.

Reflection Matrices

Reflection matrices provide an efficient way to perform reflections over specific lines on a plane. These matrices adjust coordinates so that each point of a figure is mirrored across a chosen line.

One of the most common reflect lines is \(y=x\), where each coordinate \((x, y)\) is swapped to \((y, x)\). The matrix for this reflection is:\[\begin{bmatrix}0 & 1 \1 & 0\end{bmatrix}\]

Reflection matrices are powerful tools in linear transformations, as they simplify computations involving symmetry. In our exercise, following the initial horizontal stretch, the reflection matrix swaps the "stretched" x and y coordinates. When combined in matrix multiplication with other transformations, reflections can result in complex and intricate modifications to the shape or object being transformed, demonstrating the versatility and importance of understanding these matrices.