Problem 2 - Differential Equations and Linear Algebra

Question

For the transformation of \(\mathbb{R}^{2}\) with the given matrix, sketch the transform of the square with vertices \((1,1),(2,1),(2,2),\) and (1,2). $$A=\left[\begin{array}{rr}0 & 1 \\\\-1 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The transformed square has vertices \( (1,-1), (1,-2), (2,-2), \) and \( (2,-1) \).

Step 1: Identify the square vertices

We have the square vertices: (1,1), (2,1), (2,2), and (1,2).

Step 2: Write the transformation matrix A

The given matrix A is: \[A = \left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right]\]

Step 3: Transform the vertices of the square

To transform the square vertices, we need to multiply each vertex with the transformation matrix A. For vertex (1,1): \[ \left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right] \left[\begin{array}{r} 1 \\ 1 \end{array}\right] = \left[\begin{array}{r} 1 \\ -1 \end{array}\right] \] For vertex (2,1): \[ \left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right] \left[\begin{array}{r} 2 \\ 1 \end{array}\right] = \left[\begin{array}{r} 1 \\ -2 \end{array}\right] \] For vertex (2,2): \[ \left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right] \left[\begin{array}{r} 2 \\ 2 \end{array}\right] = \left[\begin{array}{r} 2 \\ -2 \end{array}\right] \] For vertex (1,2): \[ \left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right] \left[\begin{array}{r} 1 \\ 2 \end{array}\right] = \left[\begin{array}{r} 2 \\ -1 \end{array}\right] \]

Step 4: Identify the transformed vertices and sketch the transformed square

The transformed vertices are (1,-1), (1,-2), (2,-2), and (2,-1). Now, we can sketch the transformed square using these vertices. To summarize, the transformed square has vertices (1,-1), (1,-2), (2,-2), and (2,-1).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation when dealing with linear transformations. To perform matrix multiplication, each entry in the resulting matrix is calculated by taking the dot product of the corresponding row of the first matrix and the column of the second matrix.
For example, consider a 2x2 matrix:

• The first row and first column product gives the top-left entry.
• The first row and second column product gives the top-right entry, and so on.

To multiply the transformation matrix \( A = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \) by a vector \( \begin{bmatrix} x \ y \end{bmatrix} \) from our example, we calculate it as:- First row: \((0 \times x) + (1 \times y)\)- Second row: \((-1 \times x) + (0 \times y)\)
This operation results in the transformed coordinate. Matrix multiplication thus provides a consistent and efficient way to apply transformations to multiple points.

Transformation Matrix

A transformation matrix is a special matrix used to perform linear transformations such as rotations, translations, scaling, and reflections on a vector space. In our exercise, the matrix \( A = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \) is used for performing a rotation transformation.
Each type of geometric transformation has its own specific matrix. For instance:

• A rotation matrix can rotate shapes around the origin.
• Translation matrices shift a shape from one position to another without rotating it.
• Scaling matrices change the size of the shape by stretching or shrinking it.

Using a transformation matrix like \( A \), we can uniformly apply transformations across all points in a given shape, making them invaluable for efficiently performing geometric operations.

Geometric Transformations

Geometric transformations are processes that alter the position, orientation, or size of a geometric shape. By applying transformations, we can analyze how shapes change and understand their properties better. In our case, the transformation matrix \( A = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \) represents a rotation, specifically a 90-degree rotation counterclockwise.
Geometric transformations are crucial in various fields such as computer graphics, robotics, and even physics, for simulating movement and changes in structures. Common geometric transformations include:

• Rotation: Rotating a shape by a specified angle around a pivot.
• Translation: Moving a shape to a different location without rotating or resizing.
• Scaling: Adjusting the size while maintaining the shape’s proportions.

The transformation applied in the exercise shifts the square's original vertices into new positions, illustrating how mathematical tools can model and predict changes in geometry.