Problem 39 - Calculus for Biology and Medicine

Question

$$ \begin{array}{l} \text { In Problems , give a geometric interpretation of the map }\\\ \mathbf{x} \rightarrow A \mathbf{x} \text { for each given map } A . \end{array} $$ $$ A=\frac{1}{2}\left[\begin{array}{rr} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{array}\right] $$

Step-by-Step Solution

Answer

The map is a rotation by 30° counterclockwise and scaling by 1/2.

Step 1: Identify the Transformation

The given matrix \(A\) is \(\frac{1}{2}\left[\begin{array}{rr}\sqrt{3} & -1 \ 1 & \sqrt{3}\end{array}\right]\). This matrix can be broken down into a combination of scaling and rotation transformations.

Step 2: Compare with Standard Rotation Matrix

Notice that the standard rotation matrix in 2D is \(\left[\begin{array}{rr} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{array}\right]\). Here, if we compare the elements of matrix \(A\) to this form, we identify \(\cos \theta = \sqrt{3}/2\) and \(\sin \theta = 1/2\). This corresponds to a rotation of \(\theta = 30^{\circ}\) (or \(\pi/6\) radians).

Step 3: Identify the Scaling Factor

The matrix is multiplied by \(1/2\), which means that every vector \(\mathbf{x}\) will be scaled by \(\frac{1}{2}\) in addition to being rotated.

Step 4: Geometric Interpretation

The transformation \(\mathbf{x} \rightarrow A\mathbf{x}\) can be described as a rotation by \(30^{\circ}\) counterclockwise about the origin, followed by scaling all distances from the origin to half of their original length.

Key Concepts

Rotation Matrix

To grasp the concept of a Rotation Matrix, it's important to understand how it works in two-dimensional geometric transformations. A rotation matrix is used to rotate a point around the origin by a specific angle in a coordinate plane. This matrix is essential in changing the direction of a point or shape while preserving its distance from the origin.

The standard form of a 2D rotation matrix is:

• \[\begin{bmatrix} \cos \theta & -\sin \theta \\sin \theta & \cos \theta \end{bmatrix}\]

Here, \(\theta\) is the angle of rotation. If the angle is positive, the rotation is counterclockwise, and if negative, the rotation is clockwise. This matrix ensures that while the points rotate, the object's size and shape remain unchanged, as rotation is an isometric transformation, meaning it preserves angles and distances.

In our specific exercise, the matrix was extracted to identify another matrix, \( \theta = 30^{\circ} \), resulting in a counterclockwise rotation around the center of origin, revealing how this matrix manipulates points on the plane by rotating them without changing their overall dimensions.

Matrix Scaling

Matrix Scaling is a transformation method where geometric scaling is applied, either enlarging or shrinking objects in 2D space. It involves multiplying the coordinates of a point by a constant scaling factor. A scaling transformation uniformly scales, or shrinks, an object by a constant factor.

The basic scaling matrix for 2D is:

• \[\begin{bmatrix} s & 0 \ 0 & s \end{bmatrix}\]

Where \(s\) is the scaling factor. If \(s > 1\), the object will enlarge; if \(0 < s < 1\), the object will shrink; and if \(s = 1\), the object remains the same size.

In the given exercise, a factor of \(\frac{1}{2}\) was applied, indicating a reduction of the object's size by half. This scaling affects all points in the plane, making the entire shape appear smaller, but importantly, it doesn't alter the shape's orientation. Understanding matrix scaling is crucial as it allows for manipulation of object sizes within a graphical context while maintaining overall geometric integrity.

2D Transformations

2D Transformations are fundamental in changing an object's position, size, or orientation in the two-dimensional plane. They play a pivotal role in computer graphics, animations, and geometry.The types of transformations include:

• Translation - shifting an object's position.
• Rotation - turning the object around a point.
• Scaling - resizing the object.
• Reflection - flipping the object over a line.

In our example, we combined both rotation and scaling to transform points. First, a rotation by \(30^{\circ}\) was executed. This changed only the object's orientation. Then scaling by \(\frac{1}{2}\) was applied, which reduced their size.

It's essential to understand that each transformation maintains certain properties like distance or angle measures. Knowing how each transformation operates independently allows you to decipher more complex composite transformations, like the combination of a rotation and a scale in our exercise, providing a rich toolkit for navigating 2D spatial tasks.