Problem 38 - Calculus for Biology and Medicine

Question

$$ \begin{array}{l} \text { In Problems , give a geometric interpretation of the map }\\\ \mathbf{x} \rightarrow A \mathbf{x} \text { for each given map } A . \end{array} $$ $$ A=\left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right] $$

Step-by-Step Solution

Answer

The matrix \( A \) represents a 90-degree clockwise rotation around the origin.

Step 1: Understand the Map

The matrix \( A \) represents a linear transformation. The task is to give a geometric interpretation of how this matrix \( A \) transforms any vector \( \mathbf{x} \). In this case, the 2x2 matrix \( A \) is given as \( \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \).

Step 2: Analyze Matrix Properties

Examine the matrix \( A \). This matrix is a special transformation matrix known as a rotation matrix. It resembles the standard 2D rotation matrix for a 90-degree counterclockwise rotation. In general, a 2D rotation matrix is \( \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} \). For a 90-degree rotation, \( \cos(90°) = 0 \) and \( \sin(90°) = 1 \). The matrix simplifies to \( \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \). However, our matrix \( A \) is \( \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \), which indicates there is a mistake. The correct interpretation should be around rotating 90 degrees clockwise because our matrix is equal to \( -1 \times \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \).

Step 3: Geometric Interpretation

The transformation \( \mathbf{x} \rightarrow A \mathbf{x} \) corresponds to rotating the vector \( \mathbf{x} \) 90 degrees clockwise around the origin. This is because when the standard basis vectors \( \mathbf{i} \) (as \( \begin{bmatrix} 1 \ 0 \end{bmatrix} \)) and \( \mathbf{j} \) (as \( \begin{bmatrix} 0 \ 1 \end{bmatrix} \)) are transformed using \( A \), the vectors are mapped to \( \mathbf{j} \) and \( -\mathbf{i} \), respectively. This confirms a 90-degree clockwise rotation.

Key Concepts

Rotation Matrix

A rotation matrix is a special kind of transformation matrix used in linear algebra. It helps transform vectors, thereby rotating them in a certain direction around a specific point. These matrices provide a way to rotate vectors without distorting their lengths—that is, they keep the original vector's magnitude intact. Rotation matrices are a key concept in numerous fields, including computer graphics and physics, due to their ability to depict rotations in a concise and efficient manner.

To understand how these matrices work, consider the general 2D rotation matrix:

• For an angle \( \theta \), it takes the form:\[\begin{bmatrix}\cos \theta & -\sin \theta \\sin \theta & \cos \theta\end{bmatrix}\]
• This configuration effectively rotates points counterclockwise and retains the original shape of figures.

Rotation matrices, like our matrix \( A \), have determinant 1, ensuring that they do not change the area of a shape post-transformation, maintaining its structure.

Geometric Interpretation

The geometric interpretation of a transformation matrix involves visualizing how the transformation affects vectors within a coordinate plane. When we apply the matrix \( A = \begin{bmatrix} 0 & 1 \-1 & 0 \end{bmatrix} \) to a vector \( \mathbf{x} \), we rotate the vector 90 degrees clockwise about the origin.

This transformation can be visualized as follows:

• The vector originally pointing in the x-direction gets mapped to the negative y-direction.
• Similarly, a vector in the y-direction switches to the positive x-direction.

Hence, the transformation doesn't merely alter the direction of the vectors but aligns with our understanding of clockwise rotation. The angles between vectors and their orientation to the positive axes are preserved, confirming the vector's characteristics in the geometry post-transformation.

90-Degree Rotation

A 90-degree rotation is a specific type of angular transformation applied to vectors. This operation changes the orientation of vectors in the plane, often in a simple and predictable manner. With our matrix \( A \), the vectors rotate 90 degrees clockwise. This specific operation follows a series of defined steps, as seen when translating from one axis to the other.

In more detail:

• A vector located along the positive x-axis initially, after transformation, ends up along the negative y-axis.
• Likewise, a vector initially pointing along the positive y-axis will be rotated to align with the positive x-axis.

This process of transformation is akin to turning a wheel to face a new direction, demonstrated by our matrix \( A \). Notably, vectors retain their magnitude during this rotation, showcasing the efficiency and stability of this unique transformation in maintaining the overall system integrity.