Question

A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) counterclockwise rotation by an angle of \(45^{\circ}\) (b) vertical stretch by a factor of \(1 / 2\)

Short answer

The matrix is \( A = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{4} \end{pmatrix} \).

Step-by-step solution

Identify Rotation Matrix

To perform a counterclockwise rotation by an angle \(\theta = 45^{\circ}\), we use the rotation matrix \(R\):\[ R = \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \].Substituting \(\theta = 45^{\circ} = \frac{\pi}{4}\), we have \(\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\). Thus,\[ R = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \].

Identify Vertical Stretch Matrix

A vertical stretch by a factor of \(\frac{1}{2}\) is represented by the matrix \(S\):\[ S = \begin{pmatrix} 1 & 0 \ 0 & \frac{1}{2} \end{pmatrix} \].

Combine Transformations

To apply these transformations in sequence, first the rotation, then the vertical stretch, we multiply the matrices in this order: \(S \times R\).

Perform Matrix Multiplication

Multiply the matrices:\[ S \times R = \begin{pmatrix} 1 & 0 \ 0 & \frac{1}{2} \end{pmatrix} \times \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{1}{2}\frac{\sqrt{2}}{2} & \frac{1}{2}\frac{\sqrt{2}}{2} \end{pmatrix} \].Simplifying the results, we get:\[ A = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{4} \end{pmatrix} \].

Key concepts

Rotation Matrix

When we talk about rotating a figure in the Cartesian plane, we use a special tool called the rotation matrix. This matrix helps us to turn objects around a specific point, usually the origin, without changing their size or shape.
The rotation matrix for turning an angle \( \theta \) counterclockwise is given by:\[ R = \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \]For example, if you want to rotate a point by 45 degrees, you would find the cosine and sine of 45 degrees, both of which are \( \frac{\sqrt{2}}{2} \). Substituting these values into the rotation matrix formula, you get:\[ R = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \]
This formula ensures the figure rotates as intended, maintaining its original dimensions during the transformation.

Vertical Stretch

A vertical stretch transformation changes the height of a figure without affecting its width. In mathematical terms, it involves multiplying the y-coordinates of all points by a specific factor.
For a transformation matrix, a vertical stretch by a factor of \( \frac{1}{2} \) is represented by:\[ S = \begin{pmatrix} 1 & 0 \ 0 & \frac{1}{2} \end{pmatrix} \]
The number \(1\) in the first row means the x-coordinate remains unchanged, while the \(\frac{1}{2}\) in the second row stretches the y-coordinate. This effectively compresses the height of an object, making it half its original size, but leaves the width untouched.

Matrix Multiplication

Matrix multiplication is key to combining different transformations into a single matrix. This operation involves multiplying a series of matrices in a specific order, which tells which transformation happens first.
In our case, we first have the rotation matrix \(R\), followed by the vertical stretch matrix \(S\). These matrices are multiplied as follows:\( S \times R \), which determines the effect of both transformations on the Cartesian plane.
The multiplication itself involves:

• Taking the first row of the first matrix and multiplying it with the columns of the second matrix.
• Adding these products to get each element of the resulting matrix.

Let's consider:\[ S = \begin{pmatrix} 1 & 0 \ 0 & \frac{1}{2} \end{pmatrix} \quad \text{and} \quad R = \begin{pmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{pmatrix} \]When you perform the matrix multiplication, each resulting element helps in forming the transformed coordinates and thus guides you through the sequence of transformations.

Cartesian Plane Transformations

Transformations on the Cartesian plane help us to manipulate figures in a two-dimensional space. They include operations like translation, rotation, reflection, and stretching (both horizontal and vertical). Each of these uses different types of matrices depending on the required outcome.

• Rotation: Changes the orientation, making the figure turn about a point.
• Vertical Stretch: Alters the height of the figure while keeping its position intact.

Both transformations can be combined, as shown in the provided exercise. The transformations applied to the Cartesian plane can show how the figure's specific features alter and define the resizing, reshaping, or reorienting process.
Understanding these transformations using matrices makes it possible to shift or rotate multiple figures efficiently, adopting advanced mathematical techniques to solve real-world problems.