Problem 52 - Algebra for College Students

Question

Suppose that the ordered pair \((x, y)\) of a rectangular coordinate system is recorded as a \(2 \times 1\) matrix and then multiplied on the left by the matrix \(\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\). We would obtain $$ \left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{r} x \\ -y \end{array}\right] $$ The point \((x,-y)\) is an \(x\) axis reflection of the point \((x, y)\). Therefore the matrix \(\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\) performs an \(x\) axis reflection. What type of geometric transformation is performed by each of the following matrices? (a) \(\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right]\) (b) \(\left[\begin{array}{rr}-1 & 0 \\ 0 & -1\end{array}\right]\) (c) \(\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]\) (d) \(\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]\)

Step-by-Step Solution

Answer

(a) Y-axis reflection, (b) 180-degree rotation, (c) 90-degree counterclockwise rotation, (d) 90-degree clockwise rotation.

Step 1: Identify Matrix (a) Transformation

The matrix \( \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix} \) when multiplied by a vector \( \begin{bmatrix} x \ y \end{bmatrix} \) produces \( \begin{bmatrix} -x \ y \end{bmatrix} \). This transformation reflects the point across the y-axis.

Step 2: Identify Matrix (b) Transformation

The matrix \( \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \) when multiplied by a vector \( \begin{bmatrix} x \ y \end{bmatrix} \) produces \( \begin{bmatrix} -x \ -y \end{bmatrix} \). This transformation is a rotation by 180 degrees about the origin.

Step 3: Identify Matrix (c) Transformation

The matrix \( \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \) when multiplied by a vector \( \begin{bmatrix} x \ y \end{bmatrix} \) produces \( \begin{bmatrix} -y \ x \end{bmatrix} \). This transformation is a rotation by 90 degrees counterclockwise about the origin.

Step 4: Identify Matrix (d) Transformation

The matrix \( \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \) when multiplied by a vector \( \begin{bmatrix} x \ y \end{bmatrix} \) produces \( \begin{bmatrix} y \ -x \end{bmatrix} \). This transformation is a rotation by 90 degrees clockwise about the origin.

Key Concepts

Geometric Transformations

Geometric transformations involve changing the position, size, shape, or orientation of a figure in a geometric space. They help us understand how shapes and figures can be manipulated within a coordinate system. There are several types of geometric transformations, each serving a unique purpose:

• **Translation**: Moving a figure without rotating or flipping it.
• **Rotation**: Turning a figure around a fixed point.
• **Reflection**: Flipping a figure over a line to produce a mirror image.
• **Scaling**: Enlarging or reducing the size of a figure proportionally.

In the context of matrices, we often use them to perform these transformations on points or shapes in a coordinate system. By applying matrices to ordered pairs representing points, we can achieve transformations like reflection, rotation, and scaling. Understanding these concepts helps in fields such as computer graphics, robotics, and geometry. Whenever we apply a geometric transformation, it's crucial to consider the properties of the transformation and their impact. For instance, reflections preserve the size and shape of a figure but change its orientation.

Matrix Multiplication

Matrix multiplication is a powerful tool used to perform transformations in a systematic way. When dealing with transformations in geometry, we represent points as matrices. Specifically, a point \((x, y)\) is represented as a column matrix \[\begin{bmatrix} x \ y \end{bmatrix}.\]Matrix multiplication involves taking two matrices and producing another matrix as a result. For transformations, the general idea is to take a transformation matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]and multiply it with a point matrix.The steps for multiplying matrices involve:

• Multiply each element of the rows in the first matrix by the corresponding elements of the columns in the second matrix.
• Sum the products to get the element in the resulting matrix.

This way, matrix multiplication allows us to apply transformations such as reflections and rotations efficiently. For example:- A reflection across the x-axis is represented as \[\begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}.\]- A rotation by 90 degrees counterclockwise around the origin is represented as \[\begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix}.\]Using these matrices, matrix multiplication becomes an effective way for executing geometric transformations in mathematical problems and real-world applications.

Coordinate System

A coordinate system is a method for locating points in a plane or space by using numbers called coordinates. In two dimensions, we mainly use the rectangular (or Cartesian) coordinate system, where every point is defined by an ordered pair \(x, y\). This system consists of two perpendicular axes:

• **The x-axis**, which runs horizontally.
• **The y-axis**, which runs vertically.

The point \(0, 0\) is known as the origin, the central point where the two axes intersect. By using a coordinate system, we can easily locate points, plot graphs, and solve geometric problems.In the context of matrix transformations, the coordinate system is the backdrop against which these transformations occur. When applying transformations via matrices, we alter the coordinates of points to reflect movements like rotations and reflections. For example, a reflection across the y-axis transforms a point \(x, y\) into \(-x, y\), while a reflection across the x-axis transforms it into \(x, -y\). These transformations help visualize how figures behave in space and predict the outcome of applied changes. Thus, understanding the nature of a coordinate system is crucial for effectively working with transformations in geometry.